The motions of the liquid within the Earth's outer core lead to magnetic field variations together with mass distribution changes. As the core is not accessible for direct observation, our knowledge of the Earth’s liquid core dynamics only relies on indirect information sources. Mainly generated by the core dynamics, the surface geomagnetic field provides information about the variations of the fluid motion at the top of the core. The dynamic of the fluid core is also associated with mass distribution changes inside the core and produces gravitational field time fluctuations. By applying several statistical blind source separation methods to both the gravity and magnetic field time series, we investigate the common space–time variabilities. We report several robust interannual oscillations shared by the two observation sets. Among those, a common mode of around 7 years looks very significant. Whereas the nature of the driving mechanism of the coupled variability remains unclear, the spatial and temporal properties of the common signal are compatible with a core origin.

The Earth’s magnetic field has been decreasing in strength over the past centuries, being reduced by 10 % over the last 150 years

Understanding the core dynamics involves a better understanding of not only the geomagnetic field and its variations but also of other possible observables. Indeed, our knowledge of the Earth's liquid core dynamics only comes from indirect sources of information. The dynamics of the core fluid change the magnetic field; apply a heterogeneous pressure field to the core–mantle boundary (CMB) topography, deforming the inner Earth; move density heterogeneities

Analyzing such information sources in terms of core dynamics is a challenging task, as the Earth is a complex dynamic system, which implies that all those observables are sensitive to many other sources of fluctuations. In particular, the climate dynamics dominate gravity, deformation, and Earth rotation change at most places and frequencies

Separating the contributions from different sources can only be achieved by using three different methods.

When one contribution is known with sufficient precision, it can be subtracted from the total signal, allowing better detecting and characterizing the other contributions.

When two or more data sets are sensitive to the same phenomena with different transfer functions, the joint analysis of those data sets can allow for the separation of the contributions from the different phenomena.

When different contributions have different time–space signatures, statistical blind source separation (BSS) methods can be used to separate them.

Searching for core signatures in surface observation requires using long-term and global data sets, as the core dynamic signatures are expected to be interannual and large to global scale

For the gravity field, we build on the time-variable gravity fields from the GRACE and GRACE Follow-On missions. These missions allow us to retrieve monthly global gravity field from 2002 to the present with a space resolution of a few hundred kilometers. In addition, we also make use of another temporal gravity field based on the satellite laser ranging (SLR)–GRACE hybrid approach, which allows us to extend our analysis from 1992.

The data and methods applied in this study are described in Sect. 2. The separated time and spatial properties of the magnetic and gravity fields, obtained from each different analysis, are elaborately described in Sect. 3. Finally, in Sect. 4, we discuss the characteristics of the retrieved common modes with regard to the literature on core dynamics, and we conclude with the main arguments that support the thesis that these variations come from the processes of the Earth's deep interior.

There have been significant breakthroughs in our understanding of rapid changes in the geomagnetic field over the past 2 decades, mainly through the use of recent satellite measurements. The Ørsted satellite was launched in 1999, followed by the CHAMP and SAC-C satellites in 2000. With the launch of the Swarm constellation, the geomagnetic field models resulting from the mission provide new insights into Earth’s interior. Indeed, these satellite data along with measurements obtained in the worldwide geomagnetic observatory network offer the possibility of deriving various geomagnetic field models of increasing complexity and accuracy.

One of the most regularly updated main geomagnetic field models is the CHAOS series

In the following, we present results based on two geomagnetic field models. They are COV-OBS.x2

The tracking of the GRACE and GRACE-FO space gravity satellite pairs allows estimating the global Earth time-variable gravity fields starting in 2002 with a monthly resolution

Several centers have computed Earth's time-variable gravity models based on the GRACE data: the Center for Space Research (CSR, USA), the Jet Propulsion Laboratory (JPL, USA), the GeoForschungsZentrum (GFZ, DE), the Groupe de Recherche en Géodésie (GRGS, France), the Goddard Space Flight Center (GSFC, USA), and the University of Technology Gratz (TU Gratz, AU)

This paper uses the IGG-SLR gravity field model

Before applying BSS techniques to the data sets, both the magnetic and gravity fields are pre-treated in order to smooth any sub-annual dynamics and produce anomalies of the fields. The linear trend, fit by the unweighted least-squares method, is subtracted from each point time series. For the gravity field, the seasonal cycle is then removed by subtracting the average of each month

The time series is then normalized to a zero mean and a unit standard deviation by dividing each data set by its corresponding standard deviation. Furthermore, anomalies at each grid are multiplied by the square root of the cosine of its latitude to take into account the weighting of the geographical grid size. While the modes are computed with the normalized data, we denormalize them to generate a map with full amplitude.

The geophysical data sets used in this study are given as gridded – longitude

Those modes are obtained by computing the eigenvalues and eigenvectors of a covariance matrix, and the methods differ in the way this covariance matrix is built. The modes are ordered in decreasing order of the variance captured by the mode. Classically with such methods, most of the variance of the signal is captured by only a few modes. This allows for dimension reduction of the data sets by keeping only the modes that capture a significant amount of variance.

The statistical significance of the obtained modes is assessed by comparing the eigenvalues with those obtained from surrogate data sets with the same properties as the original data sets

In the PCA

Singular spectrum analysis (SSA), first introduced by

Applied to more than one time series, the so-called MSSA uses a matrix composed of lag-covariance matrices of the different series. The details of the algorithm can be found in

We then apply a Monte Carlo hypothesis test against AR(1) noise to assess the statistical significance of the eigenvalues and the robustness of the obtained oscillatory pairs

The joint SVD technique works on decomposing the cross-covariance matrix of two different data sets that vary in space and time. This enables us to identify pairs of spatial patterns that capture the largest part of the common variability in the temporal domain. Cross-covariance matrix

For each mode, the dominant period (or frequency) is estimated as that of the maximum periodogram of that temporal properties. We apply the bootstrap technique to test the significance of the spectral power of the associated period

Simulations are then performed to estimate the dominant period's uncertainty by adding normal random phases to the time series in the Fourier domain to generate the surrogates with the same properties as the original time series

Here, we focus on the results from COV-OBS.x2 and IGG-SLR. They cover longer observation and/or model periods, as required by our analysis. The results obtained using the other data sets are shown in the Appendix. To ease reading, hereafter, the COV-OBS.x2 model is called the magnetic field and IGG-SLR is mentioned as the gravity field.

As a first step, we analyze the magnetic and gravity fields in two separate individual computations using PCA and MSSA. This allows us to analyze the space–time content of each data set without over-weighting the covariant part. Note that joint SVD, by definition, cannot be used for separated analysis. We show the spatial pattern as a time correlation coefficient between the PC (or the RC for MSSA) of that mode and the field variable at the same grid point as proposed by

We have performed the PCA on the normalized SA of the magnetic field model. From the applied Monte Carlo test

PCs of the magnetic field obtained from PCA (left). The corresponding correlation coefficients between the spatial patterns associated with each PC and the magnetic field are shown on the right. The white cross marks indicate the locations where the correlation significance does not reach the 95 % level.

The second mode captures 20.6 % of the total variance, with a period of

Unlike PCA, only three components are found to be significant at the 95 % level in MSSA (Fig.

ST-EOF 6 is also found to be significant at the 95 % level. However, the pair of this component, ST-EOF 7, is only significant at the 90 % level. Together, this pair constructs a mode with a period of 5.7

In summary, two oscillatory modes appear to be robust in the magnetic field, with periods of

Similar procedures are applied to the analysis of the gravity field. The significance test in PCA leads us to keep 29 components which together capture up to 99 % of the total variance (Fig.

The first three modes (Fig.

PCs of the gravity field obtained from PCA

The fourth mode oscillation is dominated by a 7.1-year oscillation and captures 8 % of the total variance. This mode strongly correlates around South and Central America and the northern part of African close to the Gulf of Guinea, and it extends from north to south along the meridian at 100

From the Monte Carlo test, we found 17 significant modes at the 95 % level with MSSA (Fig.

ST-EOFs 10 and 11 are also significant, showing a mode with a cycle of 3.88 years. The other significant ST-EOFs do not form oscillating pairs and correspond to higher frequencies. As they do not appear in the magnetic field and considering their high frequency, we do not discuss them further in the present study.

The time variability of the 7-year oscillatory modes of the gravity field resembles to some extent those of the magnetic field (

The spatial patterns differ between the magnetic and gravity field in the 7-year mode. We will return to the details of the gravity field spatial pattern in Sect.

We proceed with joint analyses of the magnetic and gravity fields to better highlight the similarities and differences between the two fields. For the joint PCA and MSSA, we concatenate the two normalized potential field data sets into a single multivariate time series. These methods generate common expansion coefficients (PCs) of both fields and two spatial eigenvectors that are presented as correlation maps.

As in the previous section, we first test the significance of the eigenvectors (Fig.

Figure

The dominant common variability between the two fields corresponds to a long-term behavior, similar to polynomial degree 2 of time (PC1 in Fig.

The interannual variation with

The third mode exhibits a time variability of 15.52

From the joint MSSA, seven ST-EOFs are identified as significant from the Monte Carlo test (Fig.

Reconstruction of the joint field of oscillatory pairs at a period length of 7.42

An oscillatory pair with a period length of 6.1

Figure

Bi-decadal and decadal variabilities dominate the first three modes in the SVD analysis, with similar spatial patterns compared to the results of the PCA of the joint field. However, the time series length of 28 years that we use in this study limits the reliability of detecting such long-term variation, and thus further elaboration on this behavior is beyond the scope of this paper.

For the fourth mode, we find an oscillatory period of 7.1 years, with a temporal correlation coefficient

The fifth PC exhibits a dominant oscillation of

The results in the joint field analyses are consistent with the ones in the separate analysis of the magnetic and gravity fields (Sect.

Meanwhile, the oscillation at a 6-year period is detected in all analyses except in the MSSA of the gravity field. Despite the use of various types of analysis and input combinations, the space and time signatures of these modes exhibited sufficient similarities to support the validity of their detection. This suggests that the results are robust and reliable.

We used the co-analysis of magnetic and gravity fields to separate between climate-induced and internal – probably core – signatures in the gravity field data. The application of different techniques also allows us to mine for common behavior between magnetic and gravity fields and to assess the robustness of the associated principal components of the time series. The consistency of those common behaviors over different data sets (see also the Appendix) further demonstrates the robustness of those signatures and confirms the obtained time series and space patterns.

The applied analyses provide rich information about the temporal and spatial behavior of the magnetic and gravity fields. In the following, we summarize these results in two dedicated figures.

A summary of significant mode periods is displayed in Fig.

Significant mode periods from each analysis, along with the uncertainty of the estimated dominant period. The error bars show the corresponding period length error estimate (

Modes within a period range of 6.5–7.5 years are found to be significant in 20 analyses out of 24. In the following, this is called the “7-year mode” and it captures on average 13.8 % of the variance, with a maximum of 35.1 % in the PCA of the COV-OBS.x2. The amplitude evolution of the PCs detected in the separated analysis and in the SVD exhibits differences: an increase for the magnetic field (Figs.

Unlike MSSA, SVD and PCA do not favor pseudo-periodic behavior. Finding time oscillations in SVD and PCA results is thus a piece of evidence that this periodic behavior is significant in both time series.

Besides the 7-year mode, oscillations with a period

The areas defined in Fig.

Maps of the areas associated with the 7-year mode where the correlation coefficients between the magnetic field (blue) or the gravity field (red) and the obtained time PC from PCA

Figure

We note that there are still limitations in isolating the signal linked to the core dynamics that hinder us from providing a deeper or more complete analysis.

Some possible mechanisms of the dynamic core processes that can perturb the gravity field have been previously proposed: changes in the density field within the volume of the core

The results presented here are encouraging in terms of looking for information on the dynamics of the Earth's core in other data sets, such as the gravity field, which might provide ancillary input as a base to build models that enhance our understanding of the properties and dynamics of the core. Over the upcoming years, longer magnetic and gravity observations will be available, allowing for a better separation between climate- and core-induced signatures. More refined work on separating the sources of the observed magnetic and gravity fields while taking into account the physical properties might also help to better isolate and understand different components in the Earth's core system.

Significant test of PCs using a Monte Carlo-type hypothesis.

Significant test of PCs using a Monte Carlo-type hypothesis.

PCs and their corresponding spatial correlation pattern of CHAOS-7.12 obtained from PCA.

Significant test of PCs using a Monte Carlo-type hypothesis.

Significant test of PCs using a Monte Carlo-type hypothesis.

PCs and their corresponding spatial correlation pattern of GRACE CSR mascon obtained from PCA.

Significant test of PCs using a Monte Carlo-type hypothesis.

Reconstruction of the joint field of oscillatory pairs at a period length of 7.1 years. The correlation patterns of CHAOS-7.12 and IGG-SLR are given on the right side. The MSSA here uses a window length of

Scatter of areas associated with the 6-year mode where the correlation coefficients between the potential fields and the obtained time PC from PCA

All analyses were done using MSSAkit (

The geomagnetic field models were acquired from DTU's geomagnetic field data portal: COV-OBS.x2 (

ATS, OdV, and MM contributed to the design and implementation of the research, the analysis of the results, and the writing of the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

We would like to thank Nicolas Gillet and Dominique Jault for fruitful discussions. The authors would also like to thank the two anonymous reviewers for their constructive reviews. The work is performed under the framework of the GRACEFUL project of the European Research Council (grant no. 855677). This study was supported by CNES as an application of the space gravity and magnetism missions.

This research has been supported by the European Research Council H2020 (grant no. 855677) and CNES as an application of the space gravity and magnetism missions.

This paper was edited by Elias Lewi and reviewed by two anonymous referees.