Electrical conductivity models of Earth's mantle
Steven Constable and Al Duba
affiliation
Abstract
Recent convergence between laboratory studies of mineral conductivity and electromagnetic field data have provided some confidence in our estimates of mantle electrical properties. Standard Olivine 2 (SO2), a model of upper mantle conductivity dominated by dry olivine, has proved both useful and reliable. High-pressure measurements of the electrical conductivity of perovskite agree so well with long-period sounding data that the location of the 670 km discontinuity can be estimated from EM measurements with a precision approaching that of seismic methods.
Recent laboratory measurements on wadsleyite and ringwoodite show the exciting result that they are about 100 times more conductive than olivine. Predictions for conductivities of a wadsleyite/ringwoodite dominated mantle are in broad agreement with models of mantle conductivity generated from EM soundings. However, inversion of geophysical data is an imprecise art, and an infinity of models is possible. When the wadsleyite/ringwoodite model is run forward to predict global geomagnetic impedance, it can be seen that moving the dominant conductivity contrast from 670 km to 410 km is incompatible with the data. The conductivity contrast at 410 km can at most be a factor of 10, implying a mid-mantle mineral assemblage that consists of only partially connected spinel mixed with more insulating majorite and pyroxene; given the success of SO2, the latter are probably comparable to olivine in conductivity.
1. Introduction
The results from laboratory electrical conductivity measurements on wadsleyite and ringwoodite, the high pressure spinel phases of olivine, by Xu , Poe, Shankland and Rubie (1998) show that these spinel phases are two orders of magnitude more conductive than olivine under similar conditions of temperature and pressure. The authors proposed an electrical conductivity profile for Earth based on these new data, and compared it to several conductivity models generated from geomagnetic sounding data. The agreement was considered reasonable. However, the weakness of this approach is that it compares models to models, and thus is susceptible to the non-uniqueness and ambiguity inherent in generating geophysical models from data, and fails to take into account how much, or how little, the geophysical models can be altered without affecting the fit to the data. A much better approach is to compare the model generated from the laboratory data to the actual geomagnetic data, to test for compatibility. Agreement will imply that the laboratory-based model is at least a viable candidate for the true profile. Disagreement is conclusive: the laboratory model cannot be representative of Earth.
Side-bar describing global geomagnetic data
Side-bar describing SO2.
2. A model that does fits the data
This model is one of the simplest structures that fits the global sounding data, and was constructed by Constable (1993) to fit his global compilation. It allows a large jump in conductivity at 670 km depth, and includes a highly conductive core. The lower mantle (below 670 km) fits the experimental data on silicate pervoskite published by Shankland et al. (1993), but the upper mantle is more conductive than SO2. (Constable (1993) was able to force the upper mantle to follow SO2 by including a conductive surface layer.) This model also fits the European geomagnetic response of Olsen (1999), whose data is highly compatible with Constable's in the overlapping frequency band.
3. The wadsleyite/ringwoodite model doesn't fit the data
The wadsleyite/ringwoodite model of Xu et al. (1998) (in blue) agrees with the model of Constable (1993) below 670 km, because it is constrained by the Shankland et al. (1993) data and we have already seen that these agree well with the geomagnetic response. Unlike Constable's (1993) model (shown here in green for reference), it agrees with SO2 above 440 km, as Xu et al.'s laboratory results on untransformed olivine agree well with the earlier olivine data on which SO2 is based. However, in the transition zone between 440 km and 670 km the spinel model is very different, and reflects the x100 increase in conductivity over untransformed olivine that these authors observed in the multi-anvil press. The light blue shading reflects the ±0.3 orders of magnitude uncertainty estimated by Xu et al.
Although this model appears compatible with geomagnetic sounding models (not shown), because geomagnetic models tend to spread some of the conductivity jump at 670 km into the upper mantle, when one does the forward calculation to predict the geomagnetic data one sees that there is a systematic and significant misfit to the real component of the data. Even the resistive exreme barely grazes the error bars of either data set. The result is clear; the average resistivity of the upper mantle is higher than the model values.
4. Another model which fits the data
Constraining the lower mantle to follow the perovskite measurements, and the uppermost mantle to follow SO2, we can find a transition zone conductivity that is just low enough to fit the data; we have already seen that even lower ones are possible, higher ones are excluded by the data. The conductivity of this transition zone is 0.05 S/m.
5. What does the new model mean?
Because the best fit to the field data do not allow such a large increase in conductivity as indicated by the laboratory results on conductivity data for the high-pressure polymorphs of olivine alone, we are forced to consider the following possibilities:
A. The field data are incorrect.
B. The model fit is incorrect.
C. The laboratory data are incorrect.
D. Conductivity in this region is controlled by something other than
the high-pressure polymorphs of olivine.
A: The geomagnetic field data are incorect.
This seems unlikely. Constable's (1993) compilation took data from 38 independent observatory estimates distributed over the entire globe. The agreement with Olsen's (1999) analysis from 42 observatories in Europe is typical of several studies that have compared independent data to the global response. Different studies make different assumptions; for example the work of Everett and Constable^2 (in progress, presented at this meeting) uses satellite data and is robust to non-P-1-0 field field geometry.
B: The model fit is incorrect.
The geomagnetic inverse problem is indeed non-unique, but there are some model characteristics that are constrained by the data. It is conductance (conductivity times thickness, or integrated conductivity) that determines the data more than conductivity alone, and so while it is easy to slip spurious resistive layers into a model, it is harder to include too much conductance. Note that we are not misfitting only part of the frequency band- too much shallow conductance ruins the entire fit. So, we know we can make the transition zone more resistive than 0.05 S/m by adding conductance elsewhere in the model, but it is hard to make it more conductive.
We have also assumed a radially symmetric Earth. This seems reasonable given that the laboratory studies show large conductivity jumps associated with the pressure-induced phase changes, jumps that are larger than lateral variations in temperature and volatiles are likely to produce. No doubt lateral conductivity variations do exist to some extent, but, by averaging the global data sets, these should be eliminated from the response function; Constable (1993) shows that the globally averaged data are compatible with a radial model.
C: The laboratory data are incorrect.
We have already noted that the olivine conductivity measurements of Xu et al. (1998) are in good agreement with SO2 and other olivine studies. This provides a good check on experimental procedure and, in particular, control on oxygen fugacity. Electrical conductivity of olivine (and by implication, its high pressure phases) has a very small dependence on oxygen fugacity in the reducing range, e.g., oxygen fugacity < 10**-5 Pa @ 1200 C (Duba and Constable 1993). However, it has been demonstrated that, at oxygen fugacity < 10-5 Pa @ 1200 C, iron is lost from olivine and electrical conductivity decreases with time due to this loss (Schock et al., 1989; Wanamaker and Duba, 1993). Because the iron-wüstite buffer is about 2 orders of magnitude more reducing than the values at which iron loss to the electrodes and conductivity decrease have been observed in olivine, it is reasonable to assume that iron loss has affected the high-pressure conductivity measurements of the Bayreuth group to at least a small degree. Thus, we can conclude that the conductivity reported for these phases is the minimum to be expected in the Earth. A similar argument can be made for the perovskite formed from olivine, however, here the situation is somewhat more complicated in that some of the iron that was in the original olivine may have been lost to reduction before the perovskite polymorph was formed Chakraborty et al. (1999) so that the iron in the perovskite could be even less (and, hence, the electrical conductivity) than that to be expected in the Earth.
So, if the laboratory results are in error, it is in the wrong direction:
i.e. it makes the problem of fitting the field data worse, not better.
D. Conductivity in this region is controlled by something other than the high-pressure polymorphs of olivine.
This represents the most likely possibility in our opinion. The question is, what conductivity to assign this other phase? Given the success of SO2, and the fact that we know the upper mantle is only 60-70% olivine, it is reasonable to assume that the other phases have conductivities similar to untransformed olivine. Measurements on pyroxene supports this idea (Duba, Boland and Ringwood, 1973), and measurements on lherzolites too are almost identical to single crystal olivine (Duba and Constable, 1993). In practice, the mixing study presented below is pretty well independent of the conductivity of the less conductive phase.
We need to model a mixture of conductive spinel with a more resistive phase, which we have agreed to assign the same conductivity as olivine. As one can imagine, the results depend critically on the geometry of the conductive phases. Fortunately, the mathematics of binary mixing laws is a well developed business. We have taken several results from the treatise of Schmeling (1986).
The Hashin-Shtrikman upper bound (HS+) provides the limiting case when the conductive phase is maximally interconnected thoughout the rock. In this case a mere 15% spinel with the conductivity measured in the laboratory will satisfy our models by generating a transition zone with a conductivity of 0.05 S/m. The Hashin-Shtrikman lower bound (HS-) provides the limiting case when the conductive phase is maximally isolated in the rock, insulated by the second material. Now, you can hide as much as 80% spinel in the mantle and still fit the data.
Happily, these bounds accomodate any reasonable estimate of the volume fraction of spinel in the mantle, and show that the geomagnetic data and laboratory data can be made to agree. Unhappily, the mathematical bounds don't allow the electrical data to usefully constrain the composition of the mantle. However, any information about mineralogical texture (or, failing that, reasonable assumptions) can place a fairly rigorous upper bound on how much spinel must exist in the transition zone. For example, allowing interconnection between spinel crystals, or textural anisotropy (exemplified here by the red curve showing isolated spinel crystals with an aspect ratio of 10:1, which is similar to a model of variabally interconnected equant crystals), then the amount of spinel in the transition zone must be bounded below 60%.
Conclusions
The transition zone in the mantle cannot be dominated by conductive olivine spinel, as the electrical resistivity between 440 km and 670 km must be at least an order of magnitude smaller than the conductivity of spinel alone before constraints from geomagnetic sounding data are satisfied. This implies a less conductive phase occupying at least 20% by volume, and more likely about 40% of the transition zone.