Geomagnetic secular variation violating the frozen-flux condition at the core surface
© 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: 15 December 2009
Accepted: 12 August 2010
Published: 13 December 2010
We consider a method to extract the part of a given geomagnetic secular variation (SV) model that is not consistent with a frozen-flux condition. This condition is usually derived from the diffusionless radial induction equation at the core-mantle boundary (CMB), and is defined explicitly in the spatial domain: radial flux changes within closed null-flux curves at the core surface are not allowed at any instant. We study here this condition in the spherical harmonic (SH) domain, relying on the SH expansion of the diffusionless equation. SV models at a certain epoch are separated into advective and non-advective parts. The advective (resp. non-advective) part satisfies (resp. does not satisfy) the frozen-flux condition redefined in the SH domain. We show that this separation is not unique. In this work, we achieve a unique separation by assuming the orthogonality of the two parts in terms of the radial SV energy at the CMB. From the recent geomagnetic models, GRIMM and CM4, we find that the non-advective part shows up mainly in the small reverse patches of the radial magnetic field at the CMB. However, non-advective behaviors are also observed outside these patches. As far as no restriction is imposed on core flow configuration, time variations of the non-advective part are not correlated to those of the SV models. However, if the flow is restricted to be tangentially geostrophic, time variations of the SV models have to be partly non-advective. Furthermore, for this flow configuration, the secular decrease of the axial dipole has to be largely non-advective.
The subscripts r and H denote the radial and horizontal components, respectively. The above equation describes the assumption that the decadal field variations r at the core-mantle boundary (CMB) are totally attributed to the advection of the field B r by the horizontal core flow v H at the top of the core (we hereafter refer to it as ‘FF assumption’). This is indeed a good approximation of the decadal SV generation process, as long as its spatial scale is no less than ~ 103 km in both the radial and horizontal directions (Braginsky and Le Mouël, 1993).
If a given SV model contains elements violating the above conditions, they must be attributed to either or both of two other possible origins. First, the modelled SV at large spatial scales can result from the advection of unmodelled MF at small spatial scales (Eymin and Hulot, 2005; Pais and Jault, 2008). The second origin is diffusion. It has been argued that this is possibly associated with a phenomenon often referred to as flux expulsion, which arises from toroidal field and upwelling core flow below the CMB (Bloxham, 1986; Gubbins, 1996, 2007). The diffusion can even be important for the quasi-steady part of the SV at large scales in space (Love, 1999).
In this study, we propose a method to specify the part of a SV model inconsistent with the FF condition. The method is applied to given magnetic field models for recent years. We do not intend to provide a test of the FF condition, but rather probe the spatial distribution, and time evolution, of a possible diffusion contribution to the model. Indeed, it seems very difficult to perform a conclusive test against the condition, as magnetic models mapped at the CMB are subject to considerable ambiguity. Their morphology at the CMB always has errors due to model variances and truncation of unresolvable components. In particular, both MF and SV models have spectra at the CMB that do not converge, and the integral conditions are sensitive to the truncation level (Holme and Olsen, 2006). Despite this difficulty, there is practically no other way to assess the FF assumption than using or building truncated field models (Gubbins, 1984; Bloxham and Gubbins, 1986; Benton and Celaya, 1991; Constable et al., 1993; O’Brien et al., 1997; Whaler and Holme, 2007; Jackson et al., 2007). Here, we also start with existing truncated field models to extract a SV part not satisfying the FF condition. Again we stress that this part is unexplained by the advection associated with the MF and core flow below their truncation levels. It can still result from the advection associated with the unmodelled MF and core flow at small scales.
Instead of the spatial domain approach investigating the SV flux through each C i , we adopt a spherical harmonic (SH) domain approach to assess SV models. An expression corresponding to Eq. (1) can be derived in SH domain by expanding the relevant quantities B r , r and v H (all defined on the spherical surface of the CMB) in a truncated series of the spherical harmonics. The obtained equations are the observation equations often used in estimating core flow models based on magnetic models (Bloxham and Jackson, 1991; Holme, 2007). By making use of these equations, one may be able to find the part of the SV which cannot be related to the core flow for a given truncated MF. This approach requires only an algebraic procedure, without any geometric concern about the domain of integration, such as the null-flux patches C i . Moreover, the FF condition redefined in the SH domain is likely to be stronger than the Backus’ FF condition, as it turns out in this study. Another technical advantage of the SH domain approach is the facility to incorporate assumptions needed for treating the non-uniqueness issue, when separating the SV parts satisfying and not satisfying the FF condition. We thus consider the SH domain approach worth an investigation, particularly when dealing with magnetic models given in SH coefficients.
In the next section we introduce the advective and non-advective SV parts and then the essential non-uniqueness of decomposing a given SV into these parts. In Section 3 we outline the SH domain expression of the FF induction equation (1) and describe our method to specify each SV part. We present the magnetic models to be examined in Section 4, and the computation results in Section 5. Geophysical interpretation of the results are discussed in Section 6, and conclusions follow in the last section.
2. Advective and Non-advective SV Contributions and Non-uniqueness of Their Separation
Notations of the fields in the spatial domain and their radial component, as well as in the SH domain.
The above decompositions of o are not unique. There are necessary conditions only, but no sufficient conditions for o to be ad or ge. On the other hand, there are only sufficient conditions (i.e. violating the FF or FF+TG condition), but no necessary conditions for o to be na or ng. It follows that one can always have another decomposition, for example, , where and , with being an arbitrary advective SV. It is even possible that o is totally due to na or ng.
The definitions of the SV parts, the non-uniqueness issue and the orthogonal decomposition are made even clearer by introducing a linear space of the general potential SV and its linear subsets (see Appendix A).
3. SH Domain Approach
In this section we describe our method to extract the non-advective or non-geostrophic SV for certain MF and SV models given at a certain epoch. In the SH domain, we find na by making an algebraic analysis of the FF induction equation expanded in spherical harmonics. A modification of this equation allows us to find ng in the same way.
3.1 FF induction equation in the SH domain
3.2 Specifying the non-advective and non-geostrophic SVs
4. Core Field Models
We use GRIMM (Lesur et al., 2008) and CM4 (Sabaka et al., 2004) as the input field models providing the coefficients bo and o. These recent field models deserve to be investigated, as we believe that they achieve a very high accuracy, particularly in depicting the core field and its time variation. The FF condition is not imposed on either of the two models through their construction. They are both continuous in time, expanded in B-spline function, though with a different order and knot interval of the interpolating functions. One can derive snapshots of MF and SV at any epoch within their supporting time periods.
GRIMM is a model constructed with vector magnetic data from the satellite CHAMP and ground-based observatories for the period 2001.0–2006.8. For the present study, we restrict the analysis to the period 2002.0–2005.0 in order to avoid the edge parts of this model, as they can be subject to degraded robustness. The short time span does not necessarily matter, because the test is done epoch by epoch against the FF condition using snapshots of MF and SV. Even for that short duration, the model seems to show remarkable variations, possibly regarded as geomagnetic jerks. It provides SV coefficients up to degree 14, but we consider them reliable only up to SH degree 12. The SV coefficients at degrees 13 and 14 have excessive powers with reference to the trend of the SV power spectrum, and are possibly contaminated by considerable noise. Therefore, we take LMF= 12 as the truncation degree of input models of MF and SV from GRIMM.
The comprehensive model CM4 is a model for a longer period, 1960.0–2002.0. This model is estimated using magnetic data primarily from ground-based observatories, supplemented by satellite measurements. Its accuracy may not compare with that of GRIMM due to the limited quantity and quality of data, but it features the well-known geomagnetic jerks which occurred succesively around the years 1969, 1978, 1991 and 1999 (Sabaka et al., 2004; Chambodut and Mandea, 2005). In our analysis we pay particular attention to the jerks, because we think of these abrupt events as special among all other SVs with respect that they are the core’s magnetic signals observed at the shortest timescale. The core field of CM4 is represented by its coefficients up to degree 14. Unlike GRIMM, the SV model of CM4 is damped at higher degrees. Yet, we take LMF= 13 as the truncation degree of input models of MF and SV from CM4, as the MF model may be affected by the lithospheric contribution at degree 14.
In Table 2, the dimensions of the matrices A and Atg are listed for both GRIMM and CM4 models.
In this section, after the singular values of the matrices A and Atg, we present the non-advective and non-geostrophic SVs actually computed with our method. These two SV parts are extracted for the same epochs, at intervals of 0.5 year for GRIMM (2002.0–2005.0) and 1.0 year for CM4 (1960.0–2002.0).
5.1 Non-advective SV
A rather complex configuration of ( na) r is also seen in these results. We find the flux density changes with inhomogeneous distribution, or even with different signs within single patches covering a large area. For example, the maps of ( na) r for both GRIMM and CM4 indicate that the northern hemispheric patch has an outstanding area of non-zero ( na) r localized below Indonesia, where the equatorial null-flux curve is particularly winding. This could not be revealed by allowing for the Backus’ FF condition. It is seen here because the FF condition used in the present work is stronger. It reduces the degree of freedom of B.. ad by NSV - p = 288, while the Backus’ FF condition by where is the number of the null-flux patches C i (the numbers are for the case with GRIMM at 2003.5). No matter what the flow configuration is (up to degree LFL), ( o) r in the winding null-flux curve below Indonesia cannot be attributed to the advection of the given MF model, though the curve is not locally closed to give a particular integral condition. In fact, the actual flow models have a difficulty in explaining downward continued SV models in this specific region, even if they are consistent with SV observations above the Earth’s surface (Wardinski et al., 2008). The computed ( na) r thus implies that SV models can possibly contain non-advective spots at the CMB which do not violate the Backus’ FF condition (2) locally, in addition to those identified as simple flux density changes within the smaller reverse patches.
5.2 Non-geostrophic SV
The -component of ng shows a remarkable fit to that of o. This indicates that the secular decay of the axial dipole intensity is mostly inconsistent with the FF+TG condition at the CMB, forming the part of observed SV that is unpredictable by the TG flow. Indeed, TG flow inversions tend to result in models that underfit the coefficient (Jackson, 1997). Note that these TG flow models can still explain the axial dipole decay to within its error, though our analysis indicates the significant incompatibility of the decay and the FF+TG condition. This comes from the fact that the inversions simply minimize SV misfit only up to the truncation degree of a given SV model, while we consider the FF+TG condition, including all possible SV prediction at the CMB.
5.3 Robustness of results
Here, we address questions regarding the robustness of our results before interpreting them in the next section. While no ambiguity has been discovered in selecting p at any epoch and, consequently, in the number of the SV coefficient bases u j forming na or ng, several factors still remain in our approach that can subsequently alter our results, including the definition of the scalar product in the spatial domain and the truncation degree LMF of input MF and SV models. Indeed, the results in the previous subsections can be flawed by artifacts, if the assumptions for the choices of the scalar product or the truncation degree are inappropriate for the true core. It is therefore worth investigating alternative choices to identify robust or unrealistic features of the various results.
The SV coefficients have been defined such that the two squared norms, T and , are both equal to the energy of r integrated over the core surface r = c, according to our choice of the scalar product (7). Mathematically, it is possible to take any other kind of scalar quantity for . For example, one may take as an alternative definition for Eq. (7), in which case represents the total field energy of integrated over r = c. We have checked, nevertheless, that the results are not significantly changed; the non-advective or non-geostrophic SV are still very similar to those shown in the previous subsections. This is simply because the new weight matrix , to be used instead of Eq. (10), has elements that differ only by a factor of 1.5 to 2.0 from those given by Eq. (10). We have also made a test with the heat norm using (Gubbins, 1975; Bloxham and Jackson, 1992). The subsequent non-advective SV still shows no siginificant differences, though it has slightly higher (resp. lower) powers at low (resp. high) SH degrees.
Figures 10(a) and 11(a) show three results of non-advective SV obtained using three different sets of MF and SV consisting of CM4 and synthetic coefficients up to degree LMF= 18. Their time variations are not really correlated with the geomagnetic jerks, but have amplitudes larger than those of and . It is implied that advection on the unresolved scales may contribute to the non-advective SV. Furthermore, the increased amplitude of is not just due to the energy of increased by adding the synthetic SV. We linearly decompose into its parts, each resulting from the original CM4 SV (up to SH degree 13) and the synthetic SV , to find that the increased amplitude is associated with both parts of . This means that even when the SV model only up to degree 13 is used as an input SV, the amplitude of non-advective SV is increased by the complexity of the radial field map of at the CMB due to an inclusion of synthetic MF. This seems in contrast with the findings of Gillet et al. (2009), who investigate the source of FF violation using CM4 MF and SV up to degree 13 and synthetic MF at higher degrees. They discuss that the small-scale MF is not the main cause of the FF violation. Their argument is derived from the analysis based on the flow models which are built using a damping of the higher degree components. Our incompatible statement might come from the absence of limitation with regard to the flow. We examine only field models; even flows with physically unacceptable behaviors are considered in our analysis. In addition, our FF condition is stronger in reducing the degree of freedom of ad than the ones they study, i.e. the temporal change of flux in a single patch and the temporal change of the total unsigned flux.
We also compute non-advective SVs obtained with LMF = 26, and again confirm that their time variations, which are all out of phase with the geomagnetic jerks, have amplitudes similar to those of as presented in Figs. 10(a) and 11(a). Gillet et al. (2009) introduce an exponential law fit to S(l) (as opposed to the power law fit by Lesur et al. (2008)) for extraporating the time constant to the unresolved scales. In such a case, both S(l) and the SV power spectrum at the CMB increase even more rapidly with l above degree 13, with the SV power at these degrees becoming significantly larger than the power of the synthesized SV in this study. Subsequent na may even have a larger amplitude varying with shorter timescales. It seems unlikely, nevertheless, that na happens to gain a correlation with the successive geomagnetic jerks. We can at least regard it as robust that na does not have to be correlated with the geomagnetic jerks.
On the other hand, non-geostrophic SVs obtained with LMF = 18 evolve in phase with o (Figs. 10(b) and 11(b)). They clearly exhibit the features of the geomagnetic jerks. Furthermore, there is again a persistent agreement of the components of and (Fig. 11(b)). These findings are also seen in obtained with LMF = 26. This indicates that the typical behavior of ng at the lowest SH degrees is not so sensitive to the unresolved small-scale interactions between the MF and flow, as far as the R(l) and S(l) used here are considered, and MF and SV models have the same truncation degree (at least up to 26). We conclude that the characteristics of ng shown in the previous subsection appear to be robust, in a qualitative sense.
The non-advective and non-geostrophic SVs extracted from the GRIMM and CM4 models can be indicative of some different core processes. As the potential contribution of unresolved scale advection to the large-scale SV is not negligible (Pais and Jault, 2008), this contribution to the computed non-advective and non-geostrophic SVs cannot be ruled out. Nevertheless, we are here most interested in discussing the possible diffusion contribution to our computation results. The following discussions rely on na and ng computed with LMF = 13, whose time variations and spatial distributions are presented in Subsections 5.1 and 5.2.
The computed na suggests that the non-advective SV is localized and makes up only a minor fraction of the whole SV. The local intensive spots of ( na) r at the CMB, evolving gradually as the null-flux curves nearby change their configuration, may indicate the area of diffusion associated with the flux expulsion (Bloxham, 1986; Gubbins, 2007). The rapid fluctuations of the large-scale components (excluding the axial dipole component) of o, such as the global geomagnetic jerks, are not attributed to those of na, which vary rather slowly in time (Fig. 4). This is more or less consistent with the arguments of steady diffusion (Voorhies, 1993; Love, 1999) as well as with the arguments for the primary contribution of the advection to the geomagnetic jerks (Bloxham et al., 2002; Olsen and Mandea, 2008; Wardinski et al., 2008).
The axial dipole component of na behaves in a different way than other components of na at low SH degrees, exhibiting relatively rapid variations associated with the amplitude of ~20 nT/yr. This is in contrast with the same component of o representing the secular decay of the axial dipole intensity. The decay is at least of centennial timescales (Finlay, 2008), which apparently applies to the arguments of steady diffusion. However, considering the FF condition alone does not necessarily require that the secular decay be totally due to diffusion (at least for the period of CM4). The decay may be due to both the diffusive process, i.e. the growth of reverse patches in the Southern Hemisphere, and the advective process, i.e. the poleward migration of the reverse patches (Gubbins et al., 2006).
The computed ng suggests that the input models o are poorly consistent with the FF+TG condition (as shown in Fig. 6), indicating that ( o) r substantially involves ( ng) r throughout the CMB. The non-geostrophic SV can be due to the advection by the ageostrophic flow in the equatorial regions where TG assumption tends to fail. Most regions at mid- and high latitudes are covered by the ‘geostrophic region’ (Chulliat and Hulot, 2001), so ( ng)r should not arise from the ageostrophic flow. It then follows that, in such regions, ng more probably originates in the diffusion processes. It seems physically difficult, nevertheless, to attribute the computed ng entirely to diffusion. According to the time-series map of ( ng) r over the CMB, its local intensive patterns do not hold for a long duration. They vary on timescales no longer than a decade, possibly in correspondence to the geomagnetic jerk occurences. If these rapid processes are due to diffusion, the field fluctuations should be generated in a thin region close to the core surface, with a thickness equivalent to the skin depth of the order ~104 m. Braginsky and Le Mouël (1993) and Jault and Le Mouël (1994) have examined the effect of such a thin layer in which the flow is driven by a dynamics distinct from that within the underlying volume of the core and claimed that a significant diffusion occurs in response to fluctuations of horizontal flow therein. They have also argued that the resulting SV can still be consistent with the FF condition, if the flow at the very surface of the core is replaced by an averaged flow in the layer. This type of diffusion could thus contribute to ge. It is unlikely that the rapidly fluctuating ng is explained by the diffusion of the flux expulsion type, unless a turbulence with significant kinetic and magnetic energies is assumed in the thin layer.
Unlike other components of ng, its axial dipole component does not fluctuate significantly in time. In fact, it agrees very well with the secular decay of the axial dipole (Fig. 8). If the decay arises from the poleward advection of reverse patches (Gubbins et al., 2006), then it has to be caused by the ageostrophic flow. It is not plausible, however, that the ageostrophic flow prevails at higher latitudes. We therefore would attribute the secular decay to the growth of reverse patches, rather than to the poleward migration of the patches.
Our study has focused on investigating the given geomagnetic models, but there is still a possibility of modifying them so that the FF and TG assumptions may hold thoughout the CMB. Once a SV model is allowed to have a certain amount of power due to the non-zero coefficients above degree LMF, there is little difficulty to render the SV model subject to the FF+TG condition. This has already been indicated by and obtained with the scalar product (12). Because of their insignificance at all degrees at the Earth’s surface or at 400 km altitude (r = a') (see Fig. 9 for the case with at r = a'), the residual parts and fit to o very well there, each still satisfying the FF and FF+TG conditions at the CMB. As a matter of fact, the Gauss coefficients of and are almost identical to those of o. The time-averaged rms magnitude of , i.e. rms misfit between and o, from GRIMM at r = a' is only as much as 6.0 × 10-4 nT yr-1. This is by far smaller than the accuracy of GRIMM. Of course, is an extreme example of the geostrophic SV, having the unrealistic spectrum with very high powers concentrated at degrees above LMF. Nevertheless, one may be able to create a magnetic model consisting exclusively of the geostrophic SV, by allowing for the ‘unknown SV’ above its truncation degree of the given models, which is associated with significant power at r = c, but observationally insignificant above the Earth’s surface.
With the SH domain approach, a snapshot of the part of SV violating the necessary condition of FF or FF+TG assumption can be built over the CMB. We have avoided the non-uniqueness in decomposing the SV into two parts violating and satisfying the necessary condition, by assuming the orthogonality of the two parts in terms of radial SV energy integrated over the CMB. This approach is advantageous in that geometric and topologic constraints, such as number and morphology of the null-flux curves, do not have to be considered for the necessary conditions.
We have revealed that the GRIMM and CM4 core field models, when each is truncated at the same degree for both MF and SV models, evidently involve the non-advective SV violating the FF condition and also the non-geostrophic SV violating the FF+TG condition. The radial component of the non-advective SV emerges mainly within small null-flux patches at the CMB. A local intensive area is also found in the neighborhood of the undulating null-flux curve at the magnetic equator. The time variation of the non-advective SV is not correlated with the geomagnetic jerks, indicating that these short-time events (as described by GRIMM and CM4) do not have to involve the diffusion process. In contrast with the non-advective SV, the radial component of non-geostrophic SV prevails throughout the core surface, for the two investigated models. Of particular interest is that the non-geostrophic SV shows short-term variations correlated with the geomagentic jerks. Further, the most part of the secular decay of the axial dipole field is due to the non-geostrophic SV. These findings are unlikely to depend on the truncation degree of the input field models, as long as both MF and SV models have the same truncation degree (at least up to 18) and moderate trends in their power spectra.
As the core flow is plausibly in the TG balance over a large area of the core surface, a majority of the non-geostrophic SV should not originate from the advection of radial MF by the ageostrophic flow. The diffusion is therefore a possible source of the steadily positive axial dipole component of the non-geostrophic SV, as well as its slowly evolving components which are also seen in the non-advective SV. Yet, the diffusion is practically unlikely to explain the non-geostrophic SV which fluctuates rapidly, producing a part of the observed geomagnetic jerks. This is possibly explained by the presence of fluctuating ageostrophic flow near the geographic equator, just as required for explaining the latest secular variation fluctuations (Olsen and Mandea, 2008). We note, nevertheless, that one cannot discard the possibility of appropriately modifying the SV models to meet the FF+TG condition completely. This can be achieved by allowing the unknown SV at degrees above the truncation degree of the given models to have a dominant power at the CMB.
The method of the present analysis is restricted to a given single epoch. We have examined an instantaneous SV model sequentially for each different epoch against the instantaneous necessary condition for a prescribed MF. We have not studied the MF models. Further, we have not extended our study to examine the entire span of field models at once. The method would then require a huge computation, while it is theoretically feasible if the scalar product we have defined for a certain epoch is somehow extended to involve the total time span. A practical and more comprehensive test of the observed field against the conditions would be to assess simultaneously the existence of reasonable MF, SV and flow models that are temporally continuous and compatible with the FF or FF+TG condition (Lesur et al., 2010). In constructing such a comprehensive field model, the SH domain approach would be an effective alternative to the modelling approaches directly allowing for the surface integral conditions in the spatial domain (Gubbins, 1984; Bloxham and Gubbins, 1986; Constable et al., 1993; O’Brien et al., 1997; Jackson et al., 2007).
We thank I. Wardinski for a thorough review of the manuscript during its preparation. MM work is IPGP contribution no. 3061.
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