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A candidate secular variation model for IGRF-12 based on Swarm data and inverse geodynamo modelling
- Alexandre Fournier^{1}Email author,
- Julien Aubert^{1} and
- Erwan Thébault^{1, 2}
https://doi.org/10.1186/s40623-015-0245-8
© Fournier et al.; licensee Springer. 2015
- Received: 28 January 2015
- Accepted: 8 May 2015
- Published: 27 May 2015
Abstract
In the context of the 12th release of the international geomagnetic reference field (IGRF), we present the methodology we followed to design a candidate secular variation model for years 2015–2020. An initial geomagnetic field model centered around 2014.3 is first constructed, based on Swarm magnetic measurements, for both the main field and its instantaneous secular variation. This initial model is next fed to an inverse geodynamo modelling framework in order to specify, for epoch 2014.3, the initial condition for the integration of a three-dimensional numerical dynamo model. The initialization phase combines the information contained in the initial model with that coming from the numerical dynamo model, in the form of three-dimensional multivariate statistics built from a numerical dynamo run unconstrained by data.
We study the performance of this novel approach over two recent 5-year long intervals, 2005–2010 and 2009–2014. For a forecast horizon of 5 years, shorter than the large-scale secular acceleration time scale (∼10 years), we find that it is safer to neglect the flow acceleration and to assume that the flow determined by the initialization is steady. This steady flow is used to advance the three-dimensional induction equation forward in time, with the benefit of estimating the effects of magnetic diffusion. The result of this deterministic integration between 2015.0 and 2020.0 yields our candidate average secular variation model for that time frame, which is thus centered on 2017.5.
Keywords
- Magnetic field
- Satellite magnetics
- Dynamo: theories and simulations
- Inverse theory
Background
The international geomagnetic reference field (IGRF) is a series of standard mathematical models produced by the International Association of Geomagnetism and Aeronomy (IAGA). Every 5 years, the new generation of IGRF comprises in particular a forecast of the evolution of the main geomagnetic field for the 5 years to come, in the form of a secular variation model (Hulot et al. 2015, and references therein). That secular variation component is used to compute the value of any geomagnetic element during the 5-year period of operation of the IGRF model. The computation rests on a linear interpolation of the field model from the start of the 5-year period to the time of interest. The latest version of the IGRF, hereafter referred to as the 12th generation of IGRF (IGRF-12), was released in December 2014 (Thébault et al. 2015a). Eight international teams proposed candidate secular variation models. The role of the task force appointed by IAGA was then to assess the quality of the candidate models and to propose a composite model based on those candidates (consult Thébault et al. 2015b, in this issue to see how the evaluation has been carried out for IGRF-12).
Forecasting the evolution of the main geomagnetic field is no trivial matter, since its time evolution is controlled by the complex interaction of fluid flow and magnetic field within Earth’s fluid core (e.g., Roberts and King 2013). The so-called geodynamo is a deterministic system with chaotic dynamics and consequently with a limited horizon of predictability. Estimates of this horizon are in the multidecadal range (Hulot et al. 2010; Lhuillier et al. 2011) that is several times the 5-year IGRF time scale. In addition, the geometric attenuation of the core field, going from the surface of the core to the surface of the Earth, favors its large scales over the smaller ones, effectively filtering out fast small-scale features that may potentially be detrimental to the quality of the forecast. Finally, the typical time scale for the acceleration of these large scales is on the order of 10 years (see e.g., Christensen et al. 2012), meaning that the geomagnetic evolution at 5 years is generally well approximated by a linear trend. These three points combined (multi-decadal horizon of predictability, small scales filtered out, 10-year geomagnetic acceleration time scale) explain why a linear extrapolation is rather successful over the lifetime of an IGRF release, even though unexpected sudden changes in the secular variation, in the form of geomagnetic jerks (Mandea et al. 2010), can at times substantially deteriorate the quality of the prediction.
The models of secular variation submitted to the previous generation of IGRF (Finlay et al. 2010b) were for the most part derived from time-dependent models of the main field itself. In this case, the submitted candidate secular variation was then the instantaneous secular variation at the terminal epoch of the era over which such field models were defined. Those models expressed the time dependency of the internal geomagnetic field either by means of a Taylor expansion or by resorting to splines. An exception was the candidate secular variation model proposed by Kuang et al. (Kuang et al. 2010) in which the time dependency was controlled by an underlying numerical model of the geodynamo. The arsenal of techniques designed in order to combine data with a prognostic numerical model goes by the generic name of data assimilation. Data assimilation, at the heart of numerical weather prediction (e.g., Talagrand et al. 1997), has raised growing interest in the context of terrestrial magnetism over the last decade (Fournier et al. 2010; Hulot et al. 2015).
In their study, Kuang and colleagues assimilated Gauss coefficients from a suite of nested geomagnetic field models spanning the past few millennia. Inspection of their initial results led them to modify their assimilation scheme and to incorporate a predictor-corrector algorithm to apply upon the forecast, the goal of which was to reduce the secular component of the forecast error that came from the numerical model itself (consult Kuang et al. 2010, for details).
We follow here the same philosophy of injecting physical laws, in the form of a numerical model of the geodynamo (to be described below), in order to estimate the average secular variation for the 5 years to come (2015–2020). Instead of relying on geomagnetic field models as “observations” to feed to a sequential assimilation scheme, we use here data from the Swarm constellation (Friis-Christensen et al. 2006) to construct a snapshot model of the field and its secular variation at epoch 2014.3, which we will refer to as the initial model throughout this study. This initial model is used, together with the multivariate statistics characterizing the variability of, and correlations within, the numerical dynamo model (Aubert and Fournier 2011; Fournier et al. 2011), to estimate an initial condition for the subsequent integration of that numerical dynamo model.
This is an instantaneous approach, in the sense that observations of the core field and its secular variation are fed to the numerical model at a single epoch. This strategy is motivated by two factors. First, we wish to resort to Swarm data alone, which cover a limited time span (recall that the Swarm mission was launched on 22nd November 2013). Second, we observe that, even though our sequential data assimilation technology has improved steadily over the past few years (Aubert and Fournier 2011; Fournier et al. 2013), we have not yet arrived to a point where, on interannual to decadal time scales, a sequence of observations assimilated in the past can result in an estimation of the start state for the forecast that is markedly better than that obtained with observations at a single epoch. In their current form, our sequential tools are more suited to analyze the behavior of the geomagnetic field over historical to archaeological time scales, essentially because the uncertainties affecting the corresponding measurements are rather large, which makes the assimilation of a sequence of measurements in time valuable. Dealing with a single epoch offers the possibility to assimilate the secular variation in addition to the field itself, which allows to place tighter constraints on the flow within the core, especially if the secular variation is known to high accuracy, which is the case for recent times. A snapshot initialization (Aubert 2013, 2014) followed by an integration is thus the option we retained for the IGRF. In the following, we describe our plan of action in Section “Methods.” The application of the methodology to design a candidate secular variation model for IGRF-12 is presented in Section “Results and discussion”. A summary and conclusion follow in Section “Conclusions.”
Methods
We describe in the following first how we construct the initial model for epoch 2014.3 from Swarm data and next how this initial model is used in conjunction with a numerical dynamo model to design a candidate secular variation model for IGRF-12.
Construction of the initial model for 2014.3
Data selection and weighting
We consider the Swarm satellite data (Olsen et al. 2013) for satellites A (Alpha), B (Bravo), and C (Charlie) from 26th November 2013 to 12th September 2014. We use systematically the latest version of the data and select in priority the reprocessed data (SW_RPRO_MAGx) then consider the data version 0302 running from November 2013 to 5th July 2014 then the data version 0301 up to 12th September 2014. We also select the data according to the quality flags that are defined in the Level-1b product definition document (National Space Institute, T.U.o.D. 2013) for the scalar measurements (Flags_F), for the vector measurements (Flags_B), the platform (Flags_Platform), and the satellite altitude (Flags_q). Flags_F are selected so that the Absolute Scalar Magnetometer (ASM) is in scalar or vector mode. We reject systematically data with the code 255. Acceptable vector data are those stamped with Flags_B =0 corresponding to the nominal mode of the Vector Field Magnetometer (VFM). We then keep only the data corresponding to a Flag_Platform ≤1, excluding in principle the data measured during the satellite maneuver. Lastly for Flag_q, we insist that at least two of the three satellite star cameras are in operating mode.
In addition, the Swarm scalar and vector Level-1b data are sub-sampled every 10 s, which corresponds to an along-track spacing of about 75 km. We separate the scalar and vector data into mid-latitudes (magnetic latitudes between −52° and 52°) and high latitudes (magnetic latitudes larger than 52° in absolute value). Vector data at polar latitudes are not considered. All scalar and vector data at mid-latitudes are taken in the 23:00–6:00 local time window, in order to minimize the contributions from the ionospheric S _{ q } field and to minimize the contamination by plasma bubbles (Park et al. 2013). In contrast, the scalar data in the polar regions are selected for all local times, under the condition that the Sun was at least 10° below the horizon.
Summary of data selection criteria
Magnetic | Data type | D _{ st } | Kp | Sun | Local times |
---|---|---|---|---|---|
latitude Θ | |||||
Θ≤52° | scalar and vector | ≤5 nT | ≤2° | ±10° | all |
Θ≥52° | scalar | ≤5 nT | ≤2° | ±10° | 23:00–06:00 |
We compute the magnetic field vector and scalar values at the epoch and location of the selected Swarm measurements predicted by the model of Thébault et al. (Thébault et al. 2010). These predictions are then subtracted from the measurements in order to identify strong outliers in the Swarm dataset. The presence of outliers is explained by the fact that quality flags are currently provisional and will be updated later (European Space Agency 2015). When a series of consecutive data contains outliers, the entire day is removed from the analysis. The days identified as containing consistently large outliers are the following ones: 25th and 26th March, 8th April 2014, 11th and 12th September 2014.
Statistics of the Swarm data used in this study
Data type | Number | Mean residual (nT) | rms misfit (nT) |
---|---|---|---|
ASM F mid-lat. | 307885 | 0.17 | 3.12 |
ASM F high-lat. | 197411 | -0.67 | 4.26 |
total F | 3.61 | ||
VFM B _{ r } | 307885 | -0.04 | 4.14 |
VFM B _{ θ } | 307885 | -0.14 | 6.02 |
VFM B _{ φ } | 307885 | 2.80 | 6.85 |
total B | 5.78 | ||
total | 1428951 | 5.11 |
The entries of W imply that scalar data located at the geographical equator are given a weight of 1, whereas scalar data located near the poles have a weight close to 0.5. Similar weights are applied to vector data in perfect agreement with scalar data (those which have dF=0), while vector data with dF≠0 are down-weighted according to formula (1) above.
Parameterization of the initial model
In addition, we co-estimate the static external field B _{ e } upto n=2 in the solar magnetic reference frame and assume a time-dependent component of degree n=1 parameterized by the D _{ st } index split into its external and internal contributions (Maus and Weidelt 2004; Olsen et al. 2006); consult http://www.ngdc.noaa.gov/stp/geomag/est_ist.html for the provisional indices.
Construction of the initial model; uncertainties
where A _{ i } is the design matrix (based on the calculation of the Fréchet derivative around the ith estimate) and d is the data vector of size N _{ d }. Three iterations suffice to achieve convergence. They are followed by two extra iterations to update W by an iteratively reweighted least-absolute deviation algorithm to further account for remaining outliers (e.g., Olsen et al. 2000).
Both these vectors have a size of N _{ y }.
An additional crustal covariance matrix
The main and crustal magnetic fields are not separable for the lowest degrees (n≤13) of the geomagnetic spectrum of interest here. In order to take into account the additional uncertainties on y ^{ o } due to the crustal field, we assume that, at these scales, the undesired crustal component is mostly induced by the axial dipole field. Building on the approach put forward by Thébault and Vervelidou (Thébault and Vervelidou 2015) to compute the associated crustal error covariance matrix C _{ c }, we give ourselves the possibility, in the following, to add C _{ c } to \(\mathbf {C}_{\mathbf {y}^{o}}^{\text {mf}}\) while estimating the magnetic field at the surface (and in the interior) of Earth’s core. For details on the procedure used to compute C _{ c }, we refer the reader to the study of Thébault and Vervelidou (Thébault and Vervelidou 2015).
Initialization of the geodynamo state for 2014.3
We now explain how the initial model for epoch 2014.3 can be used to define an initial condition for the integration of a numerical model of the geodynamo starting at this epoch. The overall procedure, termed inverse geodynamo modelling by Aubert (Aubert 2013), is described in detail in its latest implementation by Aubert (2014). We use this latest implementation in this work. In summary, it is a multi-step approach which combines the information coming from the observations, here in the form of y ^{ o } complemented with its error covariance matrices \(\mathbf {C}_{\mathbf {y}^{o}}^{\text {mf}}\) and \(\mathbf {C}_{\mathbf {y}^{o}}^{\text {sv}}\) (the former possibly augmented with C _{ c }), with the prior information contained in a numerical model of the geodynamo. The prior information is described by three-dimensional multivariate statistics connecting the variables defining the state of the dynamo, in our case a velocity field u _{dyn}, a magnetic field B _{dyn}, and a buoyancy field C _{dyn}. The numerical model of the geodynamo we resort to is the coupled Earth dynamo model (Aubert et al. 2013). The coupled Earth dynamo can reproduce two of the most salient features of the historical secular variation down-projected at the core-mantle boundary (e.g., Jackson and Finlay 2015), namely its hemispherical dichotomy (the strongest variations occurring in the so-called Atlantic hemisphere) and the westward drift of low-latitude features at a speed of about 15 km/year. In the coupled Earth model, we ascribe these two properties to a bottom-up control of the geomagnetic secular variation by the inner core (consult Aubert et al. 2013, for details).
From a technical standpoint, u _{dyn} and B _{dyn} are both decomposed into their poloidal and toroidal components. The resulting four fields and C _{dyn} are described using a spherical harmonic expansion in the horizontal direction, with truncation at degree and order 133. The radial dependency is treated using second-order finite differences, over a non-uniform grid comprising 160 depth levels in the fluid outer core, and 24 depth levels in the solid inner core. All in all, the number of independent state variables which define the three-dimensional state of the dynamo is close to 10^{7}. The dynamo is strongly driven, reaching a level of super-criticality which allows the magnetic Reynolds number to have a value of about 1000, within a factor of 2 of its estimate for Earth. This is a reason for assuming that this model describes fairly well the kinematics of the large-scale secular variation, which is governed by the induction equation. Not all model parameters have a proper Earth-like value, though, in particular the diffusive ones, which hinders the capabilities of the model to account for short-term processes (see below).
The three-dimensional multivariate statistics used in the following are constructed from a collection of 746 quasi-equidistant snapshots of the coupled Earth dynamo taken during a free-run (a numerical integration unconstrained by data) spanning approximately 67,000 years. Numerical integration is performed using a semi-implicit scheme and the adaptive numerical time step required for stability ranges in the free-run between 3 and 11 days. Let C _{ p } denote the covariance matrix built from this integration.
Estimate of B _{dyn}
The correlations between the poloidal field at the core surface and the poloidal and toroidal fields inside the core are statistically significant in the coupled Earth dynamo, which explains why a coherent picture of \(\widehat {\textbf {B}}_{\text {dyn}}\) is produced by the Kalman filter (see Fig. 2b, c for an illustration). Note that in the current version of the inverse geodynamo modelling framework, \(\widehat {\textbf {B}}_{\text {dyn}}\) is described up to spherical harmonic degree 30. Further details on Step 1, in particular concerning the correlations mentioned above, are provided in the Appendix.
in which μ _{0} and σ are the magnetic permeability of vacuum and the electrical conductivity of the core, respectively.
Core-surface flow
Estimate of u _{ dyn } and C _{ dyn }
The Appendix contains additional information on the correlations which enable the propagation of information from the surface of the core downwards in this step. The cylindrical-radial component of our preferred \(\widehat {\textbf {u}}_{\text {dyn}}\) is shown for illustration in Fig. 3b. As anticipated for a rapidly rotating system, the flow shows convincing signs of invariance along the direction of rotation, in the form of columns parallel to the rotation axis. The columns are partially disrupted since thermal and chemical convection in the coupled Earth dynamo is strongly driven. Further inspection of the flow shows that it is mantle bound underneath Indonesia and inner core bound underneath America. This reflects the prior information contained in C _{ p }: the boundary conditions applied to the coupled Earth dynamo favor a faster inner core growth (hence a stronger buoyancy release) underneath Indonesia.
The Gauss coefficients representation of the ASV (truncated at harmonic degree 8) can then potentially define a candidate secular variation for IGRF-12.
Results and discussion
The methodology presented in the previous section leaves in principle no room for tuning parameters, save for the scheme through which non-dimensional numerical dynamo quantities are cast into the dimensional world (for this last point, we use robust physical laws presumed to hold both in the numerical models and the Earth’s core, see e.g., Aubert et al. 2013; Fournier et al. 2011). The prior information supplied by C _{ p } is determined once-and-for-all; likewise, \(\mathbf {C}_{\mathbf {y}^{o}}^{\text {mf}}\) and \(\mathbf {C}_{\mathbf {y}^{o}}^{\text {sv}}\) are posterior statistics of the inverse problem solved to estimate \(\mathbf {y}^{o}_{\text {mf}}\) and \(\mathbf {y}^{o}_{\text {sv}}\).
We found, however, that two issues needed be addressed, connected with the unrealistically small uncertainties contained in \(\mathbf {C}_{\mathbf {y}^{o}}^{\text {sv}}\) on the one hand, and in the statistical (as opposed to dynamical) nature of \(\left (\widehat {\textbf {B}}_{\text {dyn}},\widehat {\textbf {u}}_{\text {dyn}},\widehat {C}_{\text {dyn}}\right)\) which can be detrimental to a short-term forecast, on the other hand.
Inflation of C\(_{\mathbf {y}^{o}}^{\text {sv}}\)
The formal uncertainties contained in \(\mathbf {C}_{\mathbf {y}^{o}}^{\text {sv}}\) are unrealistically small, with a root-mean-squared (rms) value of order 0.05 nT/year for the large-scale coefficients. In particular, unmodelled external field variations can cause correlated errors not accounted for in Eq. 9.
Let \(\widehat {\mathbf {y}}_{\text {sv}}\) denote our estimate of the vector of the Gauss coefficients of the secular variation, of size N _{ y }. We define the normalized misfit for the secular variation \(\mathcal {J}_{\text {sv}}\) as
The normalized secular variation misfit \(\mathcal {J}_{\text {sv}}\) obtained for different combinations of parameters during the initialization process
Inflation factor β | Crustal covariance C_{ c } | \(\mathcal {J}_{\text {sv}}\) |
---|---|---|
3.8 | yes | 2.42 |
95.7 | yes | 0.99 |
95.7 | no | 1.18 |
383.0 | yes | 0.77 |
2393.5 | yes | 0.66 |
The inflation factor β can be interpreted as a trade-off parameter that allows one to achieve a realistic fit to the data while also matching the coupled Earth dynamo statistics to an acceptable level. If the data errors were truly much smaller, of the level defined by Eq. 9, our candidate dynamo model would have to be dismissed, on the account of not being compatible with the observations. We find this scenario unlikely but cannot reject it on quantitative grounds. In order to reach a conclusion on this issue, a detailed analysis of the impact of unmodeled external field variations on the uncertainties affecting \(\mathbf {y}^{o}_{\text {sv}}\) should be undertaken in the near future; this is beyond the scope of the present study, which has to comply with the overall IGRF timing.
The steady flow assumption
one readily sees that errors in the initial \(\dot {\mathbf {u}}\) will impact the first term on the right-hand side and degrade the accuracy of the secular acceleration, at least early on in the calculation. In the case of forecasts over periods larger than 5 years (for instance 30 years), we found (working over the last decades) that a dynamical calculation based on the coupled Earth dynamo model can outperform a linear extrapolation. However, over 5 years, this never occurs. This observation is connected with the one made by Christensen et al. (2012), who demonstrated that in numerical dynamo simulations, the \(\boldsymbol {\nabla } \times \left (\dot {\mathbf {u}} \times \mathbf {B} \right)\) term is responsible for the low-degree (n≤10) secular acceleration. Errors in this term can therefore have a large impact at the surface of the Earth. A conservative option for a 5-year forecast is to get rid of \(\dot {{\mathbf u}}\) and to assume that the flow is steady. Benefits from considering the fully dynamical situation are to be expected for forecasts horizons longer than the large-scale secular acceleration time scale, which is ∼10 year according to Christensen et al. (2012), that is twice the IGRF time scale.
Forecast error (in nT) over recent 5-year periods for different forecasting strategies
2005.0–2010.0 | 2009.0–2014.0 | |
---|---|---|
Nocast | 398.3 | 438.4 |
Linear extrapolation | 66.0 | 57.5 |
Coupled Earth | 76.2 | 81.4 |
Steady flow | 62.5 | 59.3 |
Candidate model; uncertainties
The submitted candidate model with our estimate of its uncertainties
n | m | \(\dot {g}_{n}^{m}\) | \(\dot {h}_{n}^{m}\) | \( \Delta \dot {g}_{n}^{m}\) | \( \Delta \dot {h}_{n}^{m}\) |
---|---|---|---|---|---|
1 | 0 | 9.14 | 0.00 | [ −1.71 …0.51] | [ 0.00 …0.00] |
1 | 1 | 17.58 | −27.88 | [ −1.30 …0.27] | [ 0.00 …4.07] |
2 | 0 | −9.07 | 0.00 | [ −1.23 …0.53] | [ 0.00 …0.00] |
2 | 1 | −4.48 | −27.26 | [ −4.41 …1.27] | [ −0.04 …2.73] |
2 | 2 | 1.83 | −13.52 | [ −2.09 …0.12] | [ −0.74 …0.55] |
3 | 0 | 3.31 | 0.00 | [ −2.52 …0.74] | [ 0.00 …0.00] |
3 | 1 | −4.94 | 7.70 | [ −1.15 …0.13] | [ −0.77 …2.81] |
3 | 2 | −0.76 | −0.52 | [ −1.04 …0.06] | [ −1.29 …0.06] |
3 | 3 | −9.63 | 2.50 | [ −0.40 …1.97] | [ −2.96 …0.08] |
4 | 0 | 0.37 | 0.00 | [ −0.72 …0.74] | [ 0.00 …0.00] |
4 | 1 | −0.40 | −2.54 | [ −2.95 …1.71] | [ −0.96 …0.44] |
4 | 2 | −9.36 | 5.82 | [ −0.90 …2.73] | [ −1.46 …0.06] |
4 | 3 | 4.18 | 2.62 | [ −0.63 …0.46] | [ −0.39 …0.10] |
4 | 4 | −4.22 | −4.96 | [ 0.00 …1.09] | [ −0.22 …1.36] |
5 | 0 | −0.12 | 0.00 | [ −0.92 …0.95] | [ 0.00 …0.00] |
5 | 1 | 0.71 | 0.65 | [ −0.37 …0.22] | [ −0.45 …1.41] |
5 | 2 | −1.39 | 1.91 | [ −0.46 …0.47] | [ −1.15 …0.23] |
5 | 3 | −0.23 | −1.18 | [ −0.39 …0.00] | [ −0.12 …1.03] |
5 | 4 | 1.65 | 3.24 | [ −0.28 …0.09] | [ −0.72 …0.25] |
5 | 5 | 3.98 | −0.23 | [ −1.09 …0.23] | [ −0.48 …0.10] |
6 | 0 | 0.22 | 0.00 | [ −0.28 …0.78] | [ 0.00 …0.00] |
6 | 1 | −0.26 | −0.34 | [ −1.52 …0.95] | [ −0.68 …1.11] |
6 | 2 | −1.01 | −1.34 | [ −0.59 …1.14] | [ −0.60 …1.47] |
6 | 3 | 1.84 | −0.64 | [ −0.50 …0.52] | [ −0.16 …0.42] |
6 | 4 | −1.13 | 0.39 | [ −0.20 …0.05] | [ −0.02 …0.81] |
6 | 5 | 0.44 | 1.05 | [ −0.18 …0.04] | [ −0.27 …0.10] |
6 | 6 | 1.95 | 0.75 | [ −0.22 …0.11] | [ −0.33 …0.10] |
7 | 0 | 0.24 | 0.00 | [ −0.59 …0.84] | [ 0.00 …0.00] |
7 | 1 | −0.28 | 0.90 | [ −0.14 …0.38] | [ −0.37 …0.72] |
7 | 2 | −0.62 | 0.56 | [ −0.18 …0.09] | [ −0.50 …0.22] |
7 | 3 | 1.15 | −0.30 | [ −0.44 …0.16] | [ −0.28 …0.16] |
7 | 4 | 0.02 | −0.35 | [ −0.02 …0.10] | [ 0.00 …0.24] |
7 | 5 | −0.45 | −0.57 | [ −0.39 …0.42] | [ −0.27 …0.27] |
7 | 6 | −0.55 | 0.07 | [ −0.24 …0.05] | [ −0.05 …0.07] |
7 | 7 | 0.15 | 0.02 | [ −0.19 …0.10] | [ −0.32 …0.24] |
8 | 0 | 0.26 | 0.00 | [ −0.03 …0.43] | [ 0.00 …0.00] |
8 | 1 | −0.13 | −0.18 | [ −0.52 …0.27] | [ −0.70 …0.28] |
8 | 2 | −0.66 | 0.32 | [ −0.51 …0.33] | [ −0.12 …0.20] |
8 | 3 | 0.16 | −0.05 | [ −0.07 …0.41] | [ −0.23 …0.13] |
8 | 4 | −0.26 | 0.38 | [ −0.02 …0.07] | [ −0.07 …0.16] |
8 | 5 | 0.65 | −0.31 | [ −0.20 …0.02] | [ −0.09 …0.03] |
8 | 6 | 0.03 | −0.47 | [ −0.05 …0.19] | [ 0.00 …0.08] |
8 | 7 | −0.48 | 0.49 | [ −0.05 …0.19] | [ −0.07 …0.00] |
8 | 8 | 0.38 | −0.33 | [ −0.12 …0.11] | [ −0.13 …0.32] |
Conclusions
We have proposed a candidate secular variation model for the 12th release of the IGRF, whose design is new and rests on the injection of physical laws in the chain of production, in the form of multivariate statistics constructed from a free-run of the coupled Earth dynamo model. This chain is based entirely on in-house software. This candidate has been retained as a contributor to the secular variation component of IGRF-12. Its detailed assessment can be found in the study by Thébault et al. (2015b): suffice it to state here that the model is in conformity with the bulk of candidates, even though it is, to our knowledge, one of the two models which effectively evaluate the average secular variation by means of Eq. 14.
In 2020, the next generation of IGRF will allow us to examine how accurate our prediction has been, compared with the others. ‘Forecast-in-the-past’ experiments carried out over the last decades have already taught us that the 5-year IGRF lifetime is arguably too short a time scale for our approach to be highly beneficial: in part because the secular acceleration, whose low-degree component is controlled by the flow acceleration, has a time scale of 10 years (Christensen et al. 2012); in part because our technology can be improved.
This improvement can be sought on the numerical models themselves, who should ideally describe fast (interannual) core processes at work in the geomagnetic secular variation (e.g., Finlay et al. 2010a). This is not a trivial task. More within our reach are improvements on the inversion chain itself. First, its computational burden makes its actual truncation restricted to harmonic degree 30. Even though this is sufficient to take into account the unmodelled secular variation arising from the interaction of the small-scale magnetic field with the large-scale flow (Eymin and Hulot 2005), the robustness of the results reported here should nevertheless be established when the increase of compute power makes a higher truncation possible. Second, the sequential chain (Steps 1–3 in section “Initialization of the geodynamo state for 2014.3” above) is not entirely consistent, in the sense that the prior information supplied for Step i should contain the posterior information from Step i−1 (in the form of updated uncertainties on some components of the geodynamo state vector, and their Bayesian treatment). Current and future work includes improving on these aspects, along with testing the potential of the scheme to produce useful forecasts over multi-decadal periods.
Appendix: supplementary methodological information
This step is made of several substeps.
Down-projection and renormalization of y\(^{o}_{\text {mf}}\) and C\(_{\mathbf {y}^{o}}^{\text {mf}}\)
In the numerical dynamo model, the toroidal and poloidal components of B _{dyn} are discretized in the spherical shell by a set of complex-valued coefficients, \({B}^{t}_{kn m}\) and \({B}^{p}_{kn m}\), respectively; the k index refers to the radial level, which is comprised between 1 and 184, since the first 24 layers are used to discretize the solid inner core, and the remaining 160 layers are used to discretize the outer core. In addition, n and m are the spherical harmonic degree and order, respectively. Both n≤30 and m≤30 during Step 1, due to computational requirements.
where the dagger † implies conjugation and transposition.
In practice, H is rather sparse. It contains 1 entries only for those coefficients which are to be confronted with the \({B}^{po}_{n m}\) and 0 everywhere else. The action of H is therefore not implemented as a blind and naive matrix-vector product. In what follows, we will nevertheless find it useful to keep resorting to this formalism.
Statistics from the coupled Earth dynamo model
keeping in mind that N _{ e }=746 in this study (recall Section “Initialization of the geodynamo state for 2014.3”). Note that C _{ p B } does not have to be stored as such during the calculation, since (H C _{ p B })^{ † } and H C _{ p B } H ^{ † } are the sole matrices needed in practice (see below).
Magnetic Kalman filter
The (H C _{ p B })^{ † } matrix is of particular importance, since it effectively connects what is observed or probed (the large-scale component of the poloidal field at the core surface) with the toroidal and poloidal fields in the core interior.
For the poloidal coefficients, we note that the axial dipole coefficient is strongly correlated with its surface value throughout the depth of the core (blue track 1 in Fig. 7a, c). Other coefficients show weaker correlations (see the off-diagonal blue track 2 for instance). We also note that in the upper part of the core, substantial correlations are found along the diagonal, reflecting the effect of magnetic diffusion. All in all, these correlations result in the possibility of estimating the poloidal field inside the core, whose zonal component (dominated by the dipole) is shown in Fig. 2c.
The multivariate statistics contained in C _{ p B } also allows one to connect the toroidal field inside the core with the poloidal field at its surface. The relationship between these two is due to the interaction of the field with the flow. Strong correlations can be found as well throughout the core (see the red track 1, which peaks at about 0.6 near mid-depth, in Fig. 7c). Again, these correlations make it possible to produce a coherent estimate of the toroidal field inside the core, whose zonal component is shown in Fig. 2b.
Figure 7 shows in summary that noticeable correlations exist mostly within iso-m blocks, and that they are for the most part restricted to the large scales of the field. For the toroidal field, correlations within the m=1 block dominate; we interpret this as an indication of the imprint of the large-scale eccentric gyre of the coupled Earth dynamo (Aubert et al. 2013), which possesses strong m=0 and m=1 components, on the large-scale induction of the system.
Deep flow and buoyancy fields
Columnar flow yields correlations concentrated within iso-m blocks in Fig. 8a, b, an imprint of the Coriolis force influence. The imposed, m=1, hemispheric buoyancy release at the inner-core boundary (ICB), at the origin of the eccentric gyre, causes long-range correlations such as the one exhibited by the blue track 1 in Fig. 8c. In addition, nonlinear interaction between the gyre and the flow causes rather strong, off-diagonal (m±1) correlations in Fig. 8b, in the upper part of the core. The red track 1 in Fig. 8c corresponds to a correlation coefficient crossing the zero-line in signed value at about two thirds of the core depth towards the ICB (recall that it is the modulus of the correlation that is represented in this figure), meaning that the large-scale zonal flow in the lowermost part of the core is anti-correlated with its counterpart in the upper part of the core. This is consistent with the dynamical features of the coupled Earth dynamo model: torques acting on the inner core tend to entrain it eastward (as well as the fluid surrounding it), whereas the upper part of the core drifts westward by virtue of angular momentum conservation.
Declarations
Acknowledgements
We thank Ingo Wardinski and an anonymous referee for helpful comments which helped improve the manuscript. AF thanks Vincent Lesur and Gauthier Hulot for fruitful discussions on the Swarm mission. This work has been supported by the French “Agence Nationale de la Recherche” under the grant ANR-11-BS56-011 and by CNES. ESA is acknowledged for the provision of the Swarm data and for the principal investigator status granted following the Swarm Science and Validation call and selection. Numerical computations were partly performed on the S-CAPAD platform, IPGP, France (http://webpublix.ipgp.fr/rech/scp), and IDRIS HPC resources, under the allocations 2014-042122 and 2015-042122 made by GENCI. Figures were generated using the LATE X pstricks-add package and the generic mapping tools (Wessel et al. 2013); IPGP contribution number 3621.
Authors’ Affiliations
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