Paleomagnetic Secular Variation

The source of the Earth's magnetic field has been the subject of scientific study for more than 400 years (e.g., Gilbert, 1600). At present we believe that most of the field measured at the Earth's surface is of internal origin, generated by hydromagnetic dynamo action in the liquid‐iron outer core. Historic measurements of the geomagnetic field have documented its primarily dipolar spatial structure at the Earth's surface and its temporal variability, which is termed as secular variation. One notable characteristic of the Earth's magnetic field and secular variation is its full vector nature, with significant space‐time variability in both directions and intensity. Recent historic secular variation (HSV) studies (e.g., Thompson and Barraclough, 1982; Bloxham and Gubbins, 1985. Olson et al., 2002) have characterized the global pattern of short‐term secular variation and have related its variability to the core dynamo process.

Paleomagnetic studies make it clear, however, that the Earth's magnetic field has undergone a wider range of temporal and spatial variability than has been seen in historic times. Geomagnetic field polarity reversals have occurred intermittently in time (e.g., Cande and Kent, 1995) and the intervening time intervals of stable dipole polarity contain paleomagnetic secular variation (PSV) larger in amplitude and broader in frequency content than HSV. PSV studies have also documented occasional excursions (Watkins, 1976; Verosub and Banerjee, 1977), which are anomalous PSV fluctuations that may be aborted polarity reversals or represent a fundamentally different multipolar state of the geomagnetic field (Lund et al., 1998, 2001).

PSV is estimated from the paleomagnetic study of archeological materials, unconsolidated sediments, and rocks. The paleomagnetic methods used to recover PSV data are well documented (e.g., Butler, 1992. Tauxe, 1993. Merrill et al., 1998; Dunlop and Özdemir, 2001), and everyone noticed that rather different methods are normally used to recover estimates of paleomagnetic field direction and intensity. Therefore, historically, PSV directional data usually do not have associated paleointensity estimates and vice versa. However, over the last decade that tendency has finally been balanced by the development of numerous high‐resolution full‐vector PSV records.

This article surveys our current knowledge of PSV; it will provide an overview of PSV data sources, methods of PSV analysis, long‐term characteristics of PSV, and models for PSV behavior. The survey will discuss both intensity and directional variability. Special attention will be paid to the relationship between PSV and HSV, the evidence for or against long‐term stationarity of PSV, the relationship of PSV to excursions, and the characteristics of PSV that may be useful in dynamo studies.

PSV data

PSV data come from a wide variety of paleomagnetic studies that can be separated into three groups based on the type of sediment or rock measured, the degree of detail in stratigraphic sampling, and the degree of age control for each study. The three resulting PSV data groups are (1) studies of Quaternary‐aged sequences of unconsolidated sediments, lava flows, or archeological materials, which can be dated in detail by radiocarbon methods or oxygen isotope stratigraphy, and which are sampled in detail sufficient to resolve the temporal pattern of PSV variability (termed waveform information); (2) studies of older sediment or lava flow sequences that have waveform information but no detailed age control; and (3) studies of any aged rock or sediment sequences that have poor within sequence age control and no waveform information (sequential data are not serially correlated). The first type of PSV study can be used for a full spectrum of time series analyses (waveform, spectral, or statistical analyses); the second type of study can be used for waveform and statistical analyses; the third type of study is only suitable for statistical analysis.

The materials normally used for detailed paleomagnetic studies of PSV are archeological materials (kilns, fire pits, etc.), lava flows, and lake or marine sediment sequences. Each of these materials has inherent advantages and disadvantages for the accurate recording of PSV and the accumulation of paleomagnetic records from all three materials in parallel is the ideal way to establish regional patterns of PSV. Figure P9 illustrates the use of multiple PSV records derived from multiple data sources (data summarized in Constable et al., 2000) to reconstruct the paleomagnetic directional variability across Eurasia for the last 3 ka. PSV records ICE, Loch Lomand, Scotland (LLO), Lake Vukonjarvi, Finland (VUK), and Lake Baikal, Russia (BAI) are derived from lake sediment sequences, while the other six PSV records are based on archeological and lava‐flow measurements. The solid dots in Figure P9 are the actual paleomagnetic measurements, the heavy solid lines are interpolated equi‐spaced time series (PSVMOD1.0, available at http://earth.usc.edu/∼slund), and open circles in the most recent 400 years are estimated historical secular variation based on spherical harmonic analysis.

Figure P9

Comparison of paleomagnetic and historic secular variation directional data from Eurasia. Composite PSV records, summarized in Constable et al. (2000), extend from Iceland (ICE) to Japan (JPN). Solid dots are original paleomagnetic data, heavy solid lines are interpolated time‐series (PSVMOD1.0 available at http://earth.usc.edu/∼slund), and open circles in the last 400 years are estimated historical field variations based on spherical harmonic analysis.

Lava flows and archeological materials have the advantage that their natural remanent magnetization (NRM) is normally a thermoremanent magnetization (TRM). A TRM is acquired over time intervals of less than a few minutes to days by heating a material to high temperatures and cooling it in the presence of the geomagnetic field. Therefore, the TRM retains a truly “instantaneous” record of PSV. The primary disadvantage of archeological materials is their scarcity prior to about 2000 years b.p. The primary disadvantage of lava flows is the difficulty in finding sufficient radiocarbon dated flows within a region to develop a long duration, composite PSV record; only a few such studies are currently available in the whole world (e.g., Champion, 1980. Holcomb et al., 1986; Kissel and Laj, 2004).

Sediments acquire their NRM, termed as depositional or postdepositional remanent magnetization (DRM/PDRM), due to mechanical alignment of magnetic grains with the geomagnetic field while they are in the water column or in interstitial spaces just below the sediment‐water interface. The grains are subsequently locked into that orientation by grain‐grain contacts during dewatering, normally within 10–20 cm of the sediment‐water interface. The primary advantage of sediment sequences is their potential to provide continuous, high‐resolution PSV records far back in time from many sites around the world. The primary drawback to sediments, however, is the lower resolution of the DRM/PDRM recording process due to some degree of inherent smoothing of the PSV signal during remanence acquisition near the sediment‐water interface. In most high‐resolution sediment records, the smoothing interval can be estimated to be less than 100 years in duration, but further study is necessary to establish the role of smoothing in a better manner in the acquisition process of DRM/PDRM.

An added complexity associated with all PSV data is the limited extent to which they can be compared to HSV data. This difficulty in correlation occurs because (1) PSV data does not have the broad spatial (global) sampling distribution of HSV data, (2) the inherent vector resolution of PSV data (2° to 4° α95 at best) is significantly lower than the resolution of HSV data (typically 1° α95 or better), and (3) the radiometric ages associated with PSV have relatively large errors (ca. ±100 years or greater). PSV records derived from sediments have the added disadvantage, noted previously, of not recovering instantaneous estimates of secular variation due to inherent DRM/PDRM smoothing. Because of these differences, it is difficult to compare the space‐time structures of HSV and PSV data in detail, even though site measurements of PSV and HSV may agree within the limit of data resolution (e.g., Figure P9). What we can hope is that analysis of PSV data will yield spatial and/or temporal characteristics that relate, in some way, to observed HSV characteristics.

Three examples of total‐vector PSV records are shown in Figures P10‐P12. Figure P10 is a Holocene PSV record from wet lake sediments of Lake St. Croix (Lund and Banerjee, 1985; Lund and Schwartz, 1999), Figure P11 is a late‐Pleistocene PSV record from uplifted (dry) lake sediments surrounding Mono Lake (Liddicoat and Coe, 1979; Lund et al., 1985), and Figure P12 is a late‐Pleistocene PSV record recovered from deep‐sea sediments of the western North Atlantic Ocean (Lund et al., 2001, 2005).

Figure P10

Holocene full‐vector paleomagnetic secular variation record recovered from Lake St. Croix, Minnesota (USA) (Lund and Banerjee, 1985; Lund and Schwartz, 1999).

Figure P11

Late Pleistocene full‐vector paleomagnetic secular variation record recovered from uplifted lake sediments exposed at Mono Lake, California (USA) (Liddicoat and Coe, 1979; Lund et al., 1988). Note the presence of the Mono Lake excursion near 28000 radiocarbon years b.p. and indication of at least four cycles of a complex repeating vector waveform (α‐ɛ?) associated with the excursion.

Time‐series analysis of PSV

The conceptual framework for analysis of PSV data is based on the fact that HSV records are too short in duration to adequately characterize the totality of temporal field variability, while PSV records are too scattered spatially to routinely describe the prehistoric spatial field variability. We can hope, however, to identify spatial and short duration temporal components of the HSV that may relate to long‐term temporal components of the PSV. We can also attempt to improve the spatial estimates of PSV by, perhaps, appealing to analogs in HSV. Such a coordinated analysis of PSV and HSV can perhaps address a key to an unresolved question in secular variation studies: What is the mapping function between the observed spatial and temporal variations of the geomagnetic field? Only with a coherent view of the total spatiotemporal variability of the historic and prehistoric geomagnetic field can we properly evaluate models of the core dynamo process, which is the source of the field variability.

Analysis of PSV data uses a variety of time‐series and modeling techniques in order to delineate the spatial and temporal characteristics of PSV. Time‐series techniques that can be applied to PSV data will be considered in this section; modeling techniques will be considered in the next section. Time‐series techniques are classified into three broad categories: waveform analysis, spectral analysis, and statistical analysis. Each of these techniques has unique advantages for characterizing a particular type of PSV data and thereby providing a point of comparison with HSV data.

Waveform analysis

PSV records that display good serial correlation between adjacent data points (e.g., Figures P9‐P12) can be used to assess the actual time evolution of geomagnetic field variability. This time evolution can be characterized either by evaluating simple PSV features, such as maxima or minima of inclination, declination, or paleointensity, or more distinctive sets of features termed vector waveforms (Lund, 1996). One type of vector waveform that may have fundamental importance for dynamo studies is the vector‐looping pattern termed circularity (Runcorn, 1959. Skiles, 1970) often exhibited by HSV and PSV records (e.g., Bauer, 1895. Creer, 1983. Lund, 1996). Comparisons of simple features or vector waveforms may take place (1) within individual paleomagnetic records, (2) between records of different sites, and (3) between PSV and HSV records (where they overlap in time), and can be used to assess the temporal evolution of secular variation. Comparisons of the amplitudes and phase relationships of PSV features or waveforms with their spatial counterparts(?) in global maps of the present‐day field may be used to assess the spatial‐temporal mapping of secular variation. Below we discuss three different types of waveform comparison that document distinctive PSV characteristics.

The first type of waveform comparison that should be considered is between PSV records and HSV records from the same sites. For example, Figure P9 shows a transect of PSV records for the last 3 ka across Eurasia. Open circles in the interval a.d. 1600 to a.d. 2000 estimate the historic secular variation for each site based on spherical harmonic analyses. Such comparisons indicate that only the largest amplitude, longest period (∼200–400 years) HSV features can be correlated with the PSV variability (indicated by solid circles in Figure P9). This limited correlation is due to the fact that PSV records only resolve features greater than about 4° in amplitude and a few hundred years in duration. Even then, this correlation shows that high‐resolution PSV records can accurately record and extend the long‐term temporal variation only hinted at in HSV records.

The second type of comparison, between different high‐resolution PSV records from the same region (e.g., Europe or East Asia), can be used to assess the spatial extent of PSV features. For example, Figure P9 shows late Holocene PSV records from seven different sites in Europe (Iceland, ICE to CAU) spread over 70° of longitude (∼7000 km) and four different sites in East Asia (MON to Japan, JPN) spread over 30° of longitude (∼3000 km). Within each of these regions, it is readily apparent that a large number of directional features can be correlated among the records. (i.e., not to say that the records are identical, but variations in sampling rate and signal to noise ratios or errors in data acquisition and analysis probably can explain most of the differences in single records). One interpretation of these observations is that PSV features with periods longer than a few hundred years must correspond to spatial features that are observable in historic maps of the geomagnetic field and have spatial domains of several 1000 km.

A similar comparison between PSV records from different geographic regions is more problematic. For example, the comparison between European and East Asian PSV records in Figure P9 indicates that there is no simple correlation between directional records from the two regions that preserves phase relationships or long‐term trends in the directional data. Similarly, there is no simple vector pattern that can be traced all the way across Eurasia for the last 3000 years. It thus appears that straightforward PSV vector correlations break down beyond perhaps 5000 km. Thompson (1984) noted a similar scale of spatial coherence in the correlation of HSV waveforms. He determined that HSV could be broken down into about nine different regions over the Earth's surface. Within each region, HSV waveforms are broadly correlative, but between regions the patterns are significantly different (and uncorrelated except for an overall vector average approximating the axial dipole expectation). This lack of global‐scale correlation appears to be present in directional PSV records as well and indicates that there is no persistent westward (or eastward) drift of distinctive geomagnetic features over thousands of years, as has been suggested most recently by Merrill et al. (1998).

Paleointensity, on the other hand, does show clear evidence of global‐scale coherence in its variability. Comparisons of paleointensity records from around the world (e.g., Guyodo and Valet, 1999; Laj et al., 2000; Stoner et al., 2003) show clear evidence that most long‐term (>1000 years) paleointensity variability is correlatable on a global scale. For example, Figure P13 shows two high‐resolution relative paleointensity records recovered from deep‐sea sediments on almost opposite sides of the Earth (Stott et al., 2002). The records identify 30 paleointensity features that are synchronous in age (∼500 years resolution) based on oxygen isotope age determinations, including a distinctive paleointensity low associated with directional records of the Laschamp Excursion at each site. One future task of PSV studies is to rationalize how significantly different patterns of directional variability can exist around the world in the presence of such globally coherent paleointensity variability.

Figure P13

Late‐Pleistocene paleointensity comparison between sediment PSV records from the North Atlantic Ocean and western Equatorial Pacific Ocean (Stott et al., 2002). A number of distinctive and correlatable paleointensity features are noted in the two records; independent oxygen isotope stratigraphies suggest that these features are synchronous (∼500 year uncertainty).

A third type of waveform comparison can be made between PSV features or vector waveforms within individual paleomagnetic records. Such comparisons have occasionally identified distinctive vector waveforms that seem to recur every 2500 to 4000 years. The PSV record from Mono Lake, shown in Figure P11, illustrates this recurrence pattern. At least four recurrences (α‐δ) of basically the same complex waveform can be noted (Lund et al., 1988). With each recurrence, the waveform is slightly altered; however, the general pattern persists for more than 16 ka. It is likely that this distinctive waveform has evolved out of the Mono Lake excursion (waveform ɛ? in Figure P11). Similar recurring patterns have been noted in the Holocene PSV record from Lake St. Croix (Figure P10; Lund and Banerjee, 1985) in a late Pleistocene PSV record from Russia (Ekman et al., 1987), and after the Summer Lake Excursion recorded in lake sediments from Oregon (Negrini et al., 1994).

Recurring vector waveforms can also be identified by their distinctive sequences of vector looping (circularity). Such looping patterns (Runcorn, 1959) are characteristic of both PSV (e.g., Creer, 1983) and HSV (e.g., Bauer, 1895) and have been suggested (Skiles, 1970) to be an indicator of westward or eastward drift of the geomagnetic field (although there is no significant evidence of long‐term persistent drift of the field as noted above). However, the correlation between observed looping and drift is not unique (Dodson, 1979). Large amplitude loops, often associated with recurring vector waveforms, last about 1000 to 1400 years; small amplitude loops have also been noted that last about 500–800 years. Looping intervals might be used as an indicator of the simplest temporal scale for PSV coherence at individual sites.

Excursions

One type of vector waveform deserves special note—magnetic field excursions, which are anomalous PSV fluctuations defined by virtual geomagnetic poles (VGPs) located more than 45° away from the geographic pole. It is clear that excursions do occur; it is often not clear, however, what is their waveform morphology or whether some excursions are really artifacts of field‐laboratory measurement errors. There is growing evidence (e.g., Lund et al., 2001, 2005) that at least seventeen excursions have occurred in the Brunhes Epoch (0–780000 years b.p.). One distinctive element of all these excursions is that they occur within intervals of anomalously low global‐scale paleointensity (see Figures P11‐P13). Most excursion records are difficult to correlate around the world because of uncertainties in their age estimates. But, three of the most recent excursions are becoming better understood: the Mono Lake excursion (CA. 28000 years b.p.; Denham and Cox, 1971; Liddicoat and Coe, 1979; Figure P11), The Laschamp excursion (CA. 41000 years b.p.; Bonhommet and Zahringer, 1969; Figures P12 and P13) and the Blake Event (CA. 125000 years b.p.; Smith and Foster, 1969). For each of these excursions, there are now multiple independent records from around the world, which are sufficiently well dated to correlate the records and estimate that individual excursion records are synchronous to ∼500‐year resolution. There are also a number of high‐resolution excursion records (e.g., Liddicoat and Coe, 1979; Lund et al., 2005.rpar; that assess the local waveform patterns of excursions and the surrounding normal PSV. However, there are still not enough high‐resolution excursion records to be certain about their global pattern of field variability or relationship to normal PSV or magnetic field reversals.

Figure P12

Late Pleistocene full‐vector paleomagnetic secular variation record from deep‐sea piston core JPC‐14, western north Atlantic Ocean (Lund et al., 2001, 2005). Note the presence of the Laschamp excursion near ∼40000 years b.p.

Spectral analysis

Spectral analysis describes the frequency content of PSV over timescales of 10²–10⁵ years. Traditional ideas suggest that PSV should be a band limited process. That is, that the spectral power of PSV should markedly diminish beyond some cutoff period on the order of 10⁴ years. PSV records longer than the cutoff period should then be stationary in a statistical sense and have an average field vector direction that is constant through time for a given site. The axial dipole hypothesis, a cornerstone of plate tectonic reconstructions, assumes that the PSV process is stationary and that each site's average field vector (during intervals of normal polarity) satisfies the formula, tanI = 2tanλ, D = 0°, where λ is the site paleolatitude. One important goal of PSV studies is to test the long‐term validity of the axial dipole hypothesis and the notion of a stationary PSV spectrum with limited power at >10⁴ years.

Spectral analysis also describes the characteristic distribution of continuous spectral power within the overall PSV process (Barton, 1982, 1983) and identifies whether there are distinct frequency bands within the continuous spectrum that have higher than average spectral power. Knowledge of the PSV power spectrum and its potential changes in time and space is critical for better understanding the relationship between PSV and the core dynamo process that generates it.

The primary limitation in recovering detailed estimates of the PSV spectrum is the quality of age estimates associated with the PSV records. Records dated using radiocarbon methods over the last 40 ka or so may have systematic errors with respect to “true” time due to radiocarbon reservoir effects. They may also have random errors on the order of 10² years or worse due to random dating errors. Older PSV records can also be dated using oxygen isotope stratigraphies, but such age estimates have random uncertainties on the order of 10³ years.

Despite these limitations, important and convincing spectral estimates of late‐Quaternary PSV have been recovered from several regions around the world: Europe, North America, the Far East, Australia, and South America. Figure P14 shows the stacked PSV spectrum of unit‐vector (RMS of scalar inclination and declination spectra) and paleointensity results from three deep‐sea sediment PSV records in the western North Atlantic Ocean that are >50 ka in duration. These results provide a good overview of the long‐term PSV spectrum. The unit‐vector and paleointensity spectra are continuous in their power distribution, with the largest amplitude secular variation occurring at periods much greater than 10³ years (far beyond the range of HSV). Both spectra are red (power‐law relationship between frequency and power) for periods <5 ka–12 ka, and white (∼constant power as a function of frequency) for longer periods with no indication of a marked decrease in spectral power at the longest observed periods (∼50 ka). These characteristics suggest that the geomagnetic field is not stationary (band‐limited) over timescales of less than about 10⁵ years. The corner frequencies (boundaries between dominantly red and white spectra, indicated by arrows in Figure P14), are about 5 ka for the unit‐vector spectrum and about 12000 years for the paleointensity spectrum. One interpretation of the corner frequency is that the red spectrum describes the intrinsic dynamics of field variability associated with the dynamo process, while the white spectrum describes random perturbations of that process or longer‐term external forces acting on that process (e.g., Channell et al., 1998). In this scenario, the corner frequency in paleointensity of ∼12 ka defines the longest time constant of normal dynamo activity. The fact that the unit‐vector corner frequency is only ∼5 ka is due to the observation that vector variations only reflect nonaxial dipole field variability, not total field variability. It is probably not coincidental that the longest repeating vector waveforms noted in individual PSV records approach the unit‐vector corner frequency of ∼5 ka. There is also evidence of selected spectral bands in the red portion of the spectra with larger than average spectral power. That is the pattern noted in most other PSV spectra determined from shorter duration records (e.g., Barton, 1982, 1983). These spectral bands probably define the detailed pattern of dynamo activity in a region at any one time, but none of these components are probably periodic on a long‐term spatial or temporal scale and they should tend to smear out when spectra from different time intervals or spatial regions are stacked.

Figure P14

Paleointensity and unit‐vector spectra for a stack of three long‐term deep‐sea sediment PSV records from the western North Atlantic Ocean. One of the records, JPC‐14, is shown in Figure P12. Arrows indicate corner frequencies, which separate “white” and “red” portions of the spectra.

Most of these spectral characteristics have no temporal analog in HSV because of the short time span of historic measurement. One might, however, attempt to relate the long‐term PSV spectral characteristics to the present day geomagnetic field spatial spectrum. The spatial field due to core sources has a cutoff near spherical harmonic degree 12 and the spectral power decreases quickly from a maximum at harmonic degree 1 (dipole terms). Only spatial terms of about degree 6 or less have vector amplitudes large enough to be recorded in PSV., One might therefore hypothesize that the long‐term PSV spectrum must be due to temporal variations of spatial components of the present‐day field with spherical harmonic degree 5–6, under normal conditions.

Statistical analysis

The aspect of PSV that is easiest to analyze is its statistical behavior averaged over some interval of time. For this reason, statistical properties of PSV, within time windows in the order of 10⁵ or 10⁶ years, were the first PSV characteristics to be compared spatially, and even today they are the only PSV characteristics that can be easily compared on a global scale. Such comparisons provide the strongest evidence relating to possible long‐term stationarity of geomagnetic field behavior, the axial dipole hypothesis, and the potential global pattern of selected PSV characteristics. Statistical study of PSV follows two very different paths on the basis of sampling frequency and age control of the paleomagnetic measurements. In the first approach, paleomagnetic field directions in undated rock sequences are measured under the assumption that the age difference of successive rock units is large compared to the longest period of PSV (often assumed to be about 10000 years). Each data point is therefore assumed to be an independent random value picked from an assumed frequency‐band‐limited PSV process. Data sets from small regions, averaged over 10⁵ or 10⁶ years, are then statistically analyzed and compared with some global model of the expected statistical behavior. (Even if the underlying assumptions of spectral content are wrong, as they likely are, results from this type of statistical analysis should not be seriously biased if the data are truly randomly spaced in time.) The second approach is to measure well‐dated (either by radiocarbon or oxygen isotope methods) paleomagnetic sequences where the sampling interval is less than the shortest period of PSV (about 30 years). It is not often feasible to find sequences with such short time spacing, but useful information can be obtained with sample intervals up to perhaps 250 years (and even longer under special circumstances). Statistics are estimated from equispaced time‐series derived from the dated paleomagnetic records. This method permits spatial comparison of statistical parameters averaged over much shorter time intervals, on the order of 10³–10⁵ years. In such studies, it is likely that the statistics does not represent a “stationary” estimate of space‐time field variability but rather a “statistical snapshot” of an evolving space‐time field structure.

Statistical analyses of the probability distributions of both field vectors and their equivalent VGPs from single sites indicate that neither distribution is typically Fisherian (Fisher, 1953; spherical analog of the Gaussian or normal distribution). For example, Brock's (1971) results from equatorial Africa (Figure P15) show that both field vectors and VGPs tend to have somewhat elliptical distributions. Engebretson and Beck (1978) have summarized the statistical parameters normally used to characterize the shape of the probability distribution. It is probable that shape statistics vary systematically as a function of latitude (and longitude?) and future studies of shape statistics may provide important added characteristics of the Earth's long‐term PSV. As a starting point, it is worth noting that the range of vector variability (as a function of latitude) noted in historic maps of the geomagnetic field is comparable to the range of vector variability noted in individual PSV records from the late Quaternary. This suggests that, even though secular variation changes on timescales far beyond the range of historic measurements, the normal range of its spatial variability is completely present in the observed historic field. A similar observation has been made for the long‐term pattern of PSV angular dispersion by McFadden et al. (1988).

Figure P15

Statistical distribution of PSV directions and virtual geomagnetic poles (VGPs) from Equatorial Africa (Brock, 1971).

Currently, the two statistical parameters most often measured in PSV studies are the ΔI anomaly, which is the site mean inclination minus the expected axial dipole field inclination and the angular dispersion associated with a site's vector (or equivalent VGP) variability. The global pattern of the ΔI anomaly estimates how well the axial dipole hypothesis, the cornerstone of plate tectonic reconstructions, fits the actual geomagnetic field behavior. The global pattern of angular dispersion should provide some measure of the spatial pattern of intrinsic “energy” or “dynamics” in the core dynamo process. This variability may be due to differing proportions of dipole versus nondipole field variability (see summary in Merrill et al., 1998) or it could be interpreted as the relative importance of primary versus secondary family spherical harmonic components (McFadden et al., 1988), two orthogonal components of the geomagnetic field that may have fundamental relationships to dynamo theory (Roberts and Stix, 1972).

The ΔI anomaly was perhaps first quantified by Wilson in 1970 who noticed that the average paleomagnetic pole positions associated with individual geographic regions (e.g., Australia, Europe, North America) were always farther from the sampled region than the known geographic pole. This offset, termed the farsided effect, is due to paleomagnetic inclinations that are systematically more negative than their axial dipole expectation (negative ΔI anomaly), on average. McElhinny et al. (1996) have determined the global ΔI anomaly for the last 5 Ma (see Figure P16), on average, and noted that the ΔI anomaly is mostly negative with a maximum anomaly near the Equator and a latitudinal variation that is symmetric about the Equator and zonal (any site along a line of latitude will have the same magnitude of ΔI anomaly). Their analysis also indicated that ΔI anomaly has persisted with the same general pattern and magnitude for the last 5 Ma during both normal and reversed polarity. Other workers (e.g., Coupland and Van der Voo, 1980; Livermore et al., 1984; Schneider and Kent, 1990) have noted that a similar pattern of ΔI anomaly has persisted for at least the last 100 million years, but with slow variations in the maximum anomaly magnitude. Statistical studies of shorter‐term equispaced PSV time‐series (e.g., Lund, 1985; Lund et al., in review) indicate that the ΔI anomaly averaged over 10³ years (see open circles in Figure P16) and 10⁴ years (see open squares in Figure P16) also show the same general ΔI anomaly pattern noted in the longer‐term data. However, Constable and Johnson (2000) have argued that nonzonal components of the ΔI anomaly are significant and have persisted for the last several million years, with some Equatorial regions having more significant ΔI anomalies than other regions and some high‐latitude regions actually having positive ΔI anomalies. These observations all suggest that the spatial pattern of the ΔI anomaly is persistent but nonstationary in its detailed pattern over timescales greater than ∼10⁴ years. The implication of the ΔI anomaly for plate tectonic studies is that paleolatitude estimates derived from paleomagnetic studies may be in error by as much as 8° depending on time and paleolatitude. Similar analyses of declination have shown no significant deviations in declination values from 0° over the last several million years.

Figure P16

ΔI anomaly as a function of latitude for selected time intervals: solid symbols—last 5 Ma average, open circles last few thousand years average, open squares, last ∼10–20 ka average.

Analysis of PSV angular dispersions (either directions or their equivalent virtual geomagnetic poles, VGPs) has established that this parameter also displays a distinctive zonal pattern of amplitude variation with latitude. The average latitudinal pattern of VGP angular dispersion for the last 5 Ma (McFadden et al., 1988) is shown in Figure P17. Lund (1985) and Lund et al. (in review) have noted, however, that VGP angular dispersion averaged over 10³ and 10⁴ year intervals (open circles and open squares in Figure P17) using equispaced PSV time‐series is significantly lower than the 5 Ma average, and Merrill and McFadden (1988) have shown that it can vary on much longer (10⁷ year) timescales as well. A variety of parametric models, summarized in Merrill et al. (1998), have been developed to attempt to explain the observed spatial pattern of angular dispersion in an ad hoc manner. More recently, McFadden et al. (1988) were able to relate the long‐term average pattern of VGP angular dispersion to historical field observations and developed model G, based on the relative importance of primary versus secondary family magnetic field components (Roberts and Stix, 1972), to explain the latitude dependence.

Figure P17

VGP angular dispersion as a function of latitude for selected time intervals: solid symbols—last 5 Ma average, open circles last few thousand years average, open squares, last ∼10–20 ka average.

These statistical observations indicate clearly that PSV varies in a nonstationary manner over a variety of timescales, a pattern consistent with the PSV spectrum described above, and may be related to the continuing time evolution of regional PSV waveforms. One discordant note in this discussion of PSV nonstationarity is the qualitative sense that, even though PSV statistics may vary quantitatively in a nonpredictable manner over time, the PSV vector mean for any site is heavily “attracted” to its axial dipole expected direction. This may be due to the symmetry associated with the Earth's rotation, one of the driving forces of the dynamo. But, whatever be the cause, it seems prudent, perhaps, to think of PSV as “quasi‐stationary” in its behavior based on the overall sense of “attraction” to such a “fixed point” (the axial dipole expectation) in time and space.

Models of PSV

An alternative method for the analysis of PSV is to develop models for the observed field variability. Such models may be conceptual in nature with their primary purpose being to qualitatively estimate the style of variability that a potential dynamo source might generate, or the models may be more quantitative (essentially mathematical simulation models), with their primary purpose being to replicate observed PSV. The mathematical simulation models may be developed from more conceptual models, with sources that may have some basis in reality, or purely from empirical (unrealistic) inputs. A third group of models that characterize the actual hydromagnetic process that generates the Earth's core field is beyond the scope of this survey.

Conceptual models

Previously, various conceptual models have been proposed to qualitatively explain characteristic features of HSV. Three models that have been discussed most often are (1) dipole wobble, (2) westward (or eastward) drift of the total nondipole field, and (3) standing and drifting nondipole fields. These conceptual models have also been called upon to qualitatively explain specific components of PSV or to justify and physically explain specific mathematical simulation models of PSV.

Dipole wobble has been suggested as one component of HSV and PSV based on the presence of an 11.5° offset in the present day dipole field, its persistence during historic time, and the indication from paleomagnetic data that the field has an average declination of 0° over timescales of 10⁴ years or longer. Therefore, the average dipole field direction must have moved prehistorically and perhaps it has “wobbled” irregularly.

The paleomagnetic evidence for dipole wobble is problematic because of non uniqueness. PSV at a single site is really the nonaxial dipole variation (dipole wobble plus true nondipole variation) at that site. The proportion of dipole wobbles versus true nondipole contributions to PSV can only be assessed by analyzing globally distributed paleomagnetic data. Merrill and McElhinny (1983) carried out such an analysis of Northern Hemisphere archeomagnetic data for the last 2000 years and suggested that a significant dipole wobble component does exist. Dipole wobble contributions have also been estimated from the statistical analysis of angular dispersion (e.g., McFadden et al., 1988); however, the proportion of dipole wobble depends upon the proposed model of dipole wobble variability.

Westward drift has been suggested as an important element of HSV based on the observation that temporal changes in nondipole field components at the Earth's surface are due primarily to the westward drift in time of the spatial nondipole field components. The cause of westward drift has been related to differential rotation of the fluid outer core, where the field is generated, versus the overlying lithosphere. The importance of westward drift is complicated by the fact that some areas of the Earth have exhibited eastward drift and other areas have exhibited no drift at all during historic times (e.g., Thompson, 1984). Paleomagnetic evidence for westward drift comes from a variety of PSV observations, of which a few are unique. The circularity of PSV data has long been associated with westward or eastward drift of the paleomagnetic field, although other nonaxial dipole variations could produce the same effect. The recurring waveforms, noted earlier, may indicate westward (or eastward) drift of a complex nondipole waveform that changes very slowly in time compared to the time it takes for the waveform to drift entirely around the Earth (2500–4000 years). In such a model, similar waveform and spectral characteristics should be noted at all sites along a line of latitude. The Holocene waveform comparisons noted earlier, however, do not appear to be compatible with this model. Regional PSV comparisons within Europe, East Asia, or North America separately display waveform correlations that are consistent with a westward drift model; however, no simple correlation can apparently be made between the three regional data sets. Such a correlation between regional PSV records is necessary if persistent westward (or eastward) drift is a predominant aspect of PSV.

Standing nondipole field components have been proposed to improve the fit of drifting nondipole field components to the total HSV. If truly present, their origin might be related to standing components of fluid flow near the core mantle boundary caused by irregularities in the boundary conditions. The presence of standing nondipole components in the paleomagnetic record, however, is very difficult to evaluate because of problems of nonuniqueness and the uncertainties of spatial PSV behavior. To the extent that standing nondipole components might produce nonzonal components of I and D, their importance must be below the level of noise associated with parametric statistical analyses of long‐term PSV (average I, D; AI; angular dispersion), for all of these parameters are apparently zonal in their spatial distribution. However, in the study of late Quaternary PSV waveforms, standing nondipole sources have been suggested as reasonable (but nonunique) alternatives to westward drift to explain the observed waveform variability within individual paleomagnetic records.

Mathematical simulation models

Models that are more quantitative have also been applied to HSV and PSV. Spherical harmonic models separate the field into dipole and nondipole components and may include secular change coefficients for predicting short‐term temporal variations. The primary drawback to spherical harmonic models is their general lack of relevance to the underlying physical causes of the Earth's internal magnetic field. (The exception to this may be the separation of spherical harmonic coefficients into primary and secondary family field components.) Models based on the variability of multiple localized sources in the outer core have occasionally been used as alternatives to spherical harmonic models. These models, which may use a distribution of dipoles, current loops, or wave patterns in their formulation, are more appealing in that those sources may mimic that part of the core process associated with observed nondipole foci observed at the Earth's surface (e.g., Thompson, 1984).

Spherical harmonic models are hard to apply to PSV because of the inherent timing uncertainties associated with PSV data and because of the poor spatial distribution of most PSV data. Even so, several recent summaries of PSV for the last few thousand years (e.g., Hongre et al., 1998; Constable et al., 2000; Korte and Constable, in press), based on worldwide (but poorly distributed) sites, have begun to give us a low‐degree spherical harmonic view of PSV. Figure P18, for example, shows the geomagnetic field radial flux (Br) and non‐axial dipole radial flux (Br‐anomaly) at the core‐mantle boundary for two prehistoric epochs based on SHA of PSV time‐series (Constable et al., 2000). The advantage of PSV derived SHA data sets is that they can be tied to HSV derived SHA data sets and used to extend our view of the true global‐space‐time pattern of secular variation back thousands of years beyond the range of HSV. In this way, we can finally begin to address the global space‐time pattern of secular variation on scales appropriate to the dynamics of PSV and the geodynamo.

Figure P18

Models of geomagnetic field radial flux (Br) and nonaxial dipole radial flux (Br‐nad) at the core‐mantle boundary for two prehistoric epochs, 500 b.c. and 400 b.c. The models are the result of downward continuation of spherical harmonic models of PSV data from Constable et al. (2000).

Historically, time‐averaged PSV statistical parameters, such as the ΔI anomaly and vector dispersion, have been more amenable to time‐averaged spherical harmonic analysis. For example, the ΔI anomaly can be modeled by an axial dipole with added quadrupolar and octupolar components; the long‐term changes in ΔI can then be modeled as changes in the quadrupole‐octupole (or primary‐secondary family) amplitude ratio. (See Merrill and McElhinny (1983) for more detailed discussion.)

Localized dipole‐current‐loop models, with either standing or drifting sources, have been applied to individual high‐resolution PSV records, as well as to statistical PSV records. An example of a drifting radial dipole model for the Lake St. Croix PSV record (Figure P10; Lund and Banerjee, 1985) is shown in Figure P19. Two drifting radial dipoles plus an axial dipole are able to model almost all of the observed variability at Lake St. Croix for the last 9 ka. Unfortunately, more complicated standing radial dipole models could also fit the data. But, these models would require more sources in order to fit the characteristic phase relationships of the Lake St. Croix PSV data. The drifting radial dipole model for Lake St. Croix predicts similar PSV behavior for other sites on Lake St. Croix's latitude; the standing radial dipoles model will only produce regional coherence. The difficulty in correlating the Lake St. Croix PSV record with other global records outside of North America argues that a drifting radial dipole model is probably not appropriate to explain recurring waveforms. However, the complex recurrent waveforms noted in several PSV records are very difficult to model with standing sources due to the number of sources required and the detailed timing of recurrent intensity variations that each source must maintain relative to the other sources.

Figure P19

Radial‐dipole and dynamo‐wave models of the Lake St. Croix PSV record.

An alternative model of PSV, based on poleward migrating dynamo waves in the Earth's outer core, has been developed by Olson and Hagee (1987). Figure P19 shows the results of their model applied to the observed Lake St. Croix PSV. It is apparent that the poleward‐migrating dynamo‐wave model does just as good a job of fitting the observed variability as the drifting radial‐dipole model. The dynamo wave model, however, only requires regional coherence in waveform correlations, but not a global‐scale correlation. Hagee and Olson (1989) have also applied this dynamo wave model to a global set of Holocene PSV records and shown that the same general pattern of dynamo‐wave variability noted to Lake St. Croix, can also explain all other Holocene PSV around the world.