5.1. Electrical Conductivity of Mantle Minerals
[38] For electrical conductivity the equations of state for dry mantle minerals are reviewed in detail in Jones et al. [2009a], and, for small polaron conduction, are given as
based on the work of Hirsch et al. [1993] for Ol, Xu and Shankland [1999] for Opx, Xu et al. [2000] for Cpx, and Romano et al. [2006] for Gt (but with a correction applied, see Jones et al. [2009a]), where mineral X_{Fe} is the iron number (related to the magnesium number Mg# by X_{Fe}= 1 − Mg#/100; can be different for each mineral), k is Boltzmann's constant, and T is temperature in Kelvin. Note that the conductivities of the pyroxenes are not irondependent – this is because either the measurements have not been made or the laboratories are inconsistent in their results (see discussion inJones et al. [2009a]), so we have little choice but to assume iron dependency is negligible. Also, spinel is ignored as the evidence from mantle xenoliths is that its occurrence is well below 1% at the depth of interest in this paper (100 km).
[39] Fullea et al. [2011] revised the above equations taking into account newer laboratory studies and also the measurements of the effects of water in the minerals. The revised equations are of the general form
where
The first term in equation (10) describes small polaron conduction, and includes a dependency on mineral iron number X_{Fe}. The second term describes the contribution of Mg vacancies at high temperatures (and lacks X_{Fe}dependency). The third term describes proton conduction in the waterpresent case, as described in the general case byequation (1).
[40] At the conditions of interest in this study, the temperatures are sufficiently low that conduction from Mg vacancies is negligible (it only becomes important at temperatures beyond 1200°C).
5.2. Variation of Olivine Conductivity With Temperature
Table 4. Parameters for Conductivity Equations of Fullea et al. [2011]^{a}  Small Polaron Conduction  Mg Vacancies  Proton Conduction 

Karato Model  Yoshino/Poe Model 

log σ_{0,sp}  a  b  c  d  e  f  ΔV  log σ_{0i}  ΔH_{i}  ΔH_{wet} (eV)  log σ_{0}  r  ΔH_{wet} (eV)  log σ_{0}  α 


Ol  2.4–3.0  1.642  0.246  −4.85  3.259  –  –  0.68  4.73  2.31  0.87 ± 0.05  3.0 ± 0.4  0.62 ± 0.15  1.19  2.35  1.1 
Opx  2.4–3.72  1.9  −2.77  2.61  −1.09  –  –  –  –  –  0.85  2.6  0.62  1.19  2.35  1.1 
Cpx  3.25  2.075  −2.77  2.61  −1.09  –  –  –  –  –  0.85  2.6  0.62  1.19  2.35  1.1 
Gt  (2.67–3.39) + log_{10}(1–0.044P)  2.6  −15.33  80.4  −194.6  202.6  −75  20.5  40.96  20.05  00.725  20.29  00.63  10.19  20.35  10.1 
[42] Also shown in Figure 4 are the conductivity estimates for proton conduction for 100 ppm water from the three models, Karato (Figure 4, dashed black line), Yoshino (Figure 4, dashed red line) and Poe (Figure 4, dashed blue line). Comparing Figure 1 with Figure 4, proton conduction dominates over small polaron conduction for even small amount of water; <45 ppm for the Poe AV model and far less for the other two models. Thus, the differences between the polaron models and the errors introduced by adopting a particular polaron conductivity model, and also ignoring iron effects on pyroxene conductivity, are irrelevant at the water contents of interest in this study given the dominance of proton conduction.
[43] Note that for the water contents of interest, the proton conduction term has a relatively small gradient with temperature, regardless of model chosen, of approx. 0.37–0.42 log units for a 100°C change. Thus, a 50°C error in our presentday temperature estimates for either Jagersfontein (740°C) or Gibeon (850°C) will result in an error of 0.2 log units in our estimate of conductivity. This is the same order as the experimental errors of our resistivity estimates (0.20 log units for Jagersfontein and 0.09 log units for Gibeon, see below).
[44] The complete conductivity calculations in temperaturewater content (TC_{w}) space for the three proton conduction models are shown in Figure 5. For all of them, at low water contents temperature is virtually immaterial. This changes at high water contents, where there is a strong gradient with both temperature and water content for the Poe model, but for the Karato and Yoshino models conductivity becomes insensitive to actual water content, as discussed above.
5.3. Whole Rock Conductivity Calculation
[45] The parameters used for equations (10) and (11) for the four minerals are listed in Table 4. In implementing the generalized form for proton conduction, σ_{p}, in equation (11) for the different conduction models derived by the three laboratories, the following formulations were used (with parameters specified in Table 3): (1) For the Karato model, a constant value of ΔH_{wet} (i.e., α assumed to be zero) and f(C_{w}) = σ_{0}C_{w}^{r}. (2) For the Yoshino and Poe models, ΔH_{wet} = ΔH_{0} − αC_{w}^{(1/3)} and f(C_{w}) = σ_{0}C_{w} (i.e., r assumed to be one).
[46] Following the Jones et al. [2009a] approach, we calculate whole rock conductivity for the four samples and two averages using the (P,T) conditions and modal compositions from Table 3, the formulae in equations (10) and (11), and aggregate the individual minerals using secondorder, multiphase HashinShtrikman extremal bounds [Berryman, 1995], taking into account possible interfacial effects [Salje, 2007]. Simple mixing laws, such as arithmetic or geometric relations, Archie's Law, and Voigt and Reuss estimates, give firstorder variational bounds, whereas HashinShtrikman represents a member of the class of secondorder variational bounds. There exist thirdorder variational bounds [Beran, 1965; Beran and Molyneux, 1966] that have been applied in seismology [X F Xu, 2011] but not yet in electromagnetism. An additional term was added to the HashinShtrikman equations ofBerryman [1995] by Salje [2007]to explain observations that were found to lie outside the HashinShtrikman bounds. The term considers interfacial effects that are shown to scale withf(1f), where f is the volume proportion of the individual phase. For intermediate cases, where both volume and interfacial effects are important, the volume proportion f in the averaging scheme is replaced by f(1 − S) + Sf^{2}, where S ranges from 0 (no interfacial effects) to 1 (only interfacial effects).
[47] Extremal bounds are more important for electrical conductivity than for bulk moduli, as the influence of the highest conductivity phase, even if only a minor component (1%), can be dramatic if the minor phase is sufficiently well interconnected. However, olivine (Ol) and pyroxene (Opx, Cpx) minerals when dry have conductivities that are within an order of magnitude of each other at any given (P,T) conditions, and, for the mineral assemblages considered herein, Ol and Opx make up the bulk of the whole rock, over 95% in most cases, so the extremal bounds are not wide. We derive estimates using the geometric means of the upper and lower HashinShtrikman bounds with and without interfacial effects; the individual values are so close to each other that the geometric and arithmetic means are virtually the same.
[48] Our average petrologically based estimates for log_{10}(conductivity) at 100 km depth beneath Jagersfontein and Gibeon, under waterabsent, dry conditions, i.e., small polaron conduction only, are −5.30 and −4.39 log units respectively (Table 5).
Table 5. HashinShrikman Estimates of Log_{10}(Conductivity [S/m]) at 100 km Depth Beneath Jagersfontein (FRB Samples) and Gibeon (KGG Samples), Assuming Dry Conditions (No Water)^{a}Sample  T (°C)  HS^{−} (S = 0)  HS^{+} (S = 0)  HS^{−} (S = 1)  HS^{+} (S = 1)  HS GAV 


FRB983  710  −5.53  −5.20  −5.49  −5.49  −5.43 
FRB1007  755  −5.18  −4.76  −5.19  −5.16  −5.07 
FRB AV  740  −5.30  −4.96  −5.29  −5.28  −5.21 
KGG06  875  −4.34  −3.86  −4.36  −4.24  −4.20 
KGG65  825  −4.68  −4.12  −4.71  −4.59  −4.52 
KGG AV  850  −4.51  −3.99  −4.53  −4.41  −4.36 
[49] Given the observations of water in the lithosphere, we must derive the expected conductivities under “slightly damp,” conditions. None of the data in Peslier et al. [2010b]are from xenoliths sourced as shallow as 100 km nor as cold as 740°C, so we have to extrapolate from their results at higher temperatures (900°C) beneath Kimberley that show evidence for around 80 ppm water in the olivine. Given the lack of any depthdependence shown for the upper lithosphere inPeslier et al. [2010b], we assume a value of 80 ± 10 ppm later in the text, but calculate at this point conductivity for water content from 0–200 ppm in olivine.
[50] The conductivity calculations for wet conditions for olivine for the Karato, Yoshino and Poe (average) models that include the effects of proton conduction using equations (10) and (11) for temperatures of 740°C and 850°C, i.e., our Jagersfontein and Gibeon estimated ambient temperatures at 100 km, are shown as the dashed lines in Figure 6.
[51] Olivine is not the only mantle mineral, even though it comprises >70% (Table 3) of the rock mass, so consideration must be given to the contribution to proton conduction from the other mantle minerals, pyroxenes and garnet, of which Opx dominates. Water in pyroxenes is higher than in olivine, with averages of 200 ppm in Opx and 300 ppm in Cpx, and virtually zero in garnet, in the Kaapvaal peridotite xenolith samples reported in Peslier [2010, supplementary material]. A recent abstract lists a narrow range for both Opx and Cpx of 119–251 ppm and 149–398 ppm respectively [Peslier et al., 2010a], bracketing the average given above, with the intriguing suggestion of an increase in water content in pyroxenes from Liqhobong (Lesotho), through Finsch Mine, Jagersfontein, to Kimberley xenoliths. In contrast, far lower values are reported for xenoliths from the Letseng kimberlite in Lesotho [Sundvall and Stalder, 2011], from 20 ppm. As discussed above, we assume a water partitioning of 1:2:3 between the Ol, Opx and Cpx.
[52] Computing the resistivities for the composition given in Table 3 for water contents of x ppm in Ol, 2x ppm in Opx, 3x ppm in Cpx, and 0 ppm in Gt (values observed are 0–15 ppm at 4 Gpa in Peslier [2010], but given the very low volumes of these minerals we set the water contents to zero), where x varies from 0 to 200 and the increase in water in pyroxenes is assumed to be linearly related to the increase in water in olivine, yields the solid lines shown in Figure 6a for the conductivity beneath Jagersfontein with varying water content. Holding the water contents of the pyroxenes fixed at the averages of 200 ppm and 300 ppm for Opx and Cpx respectively and varying Ol only from 0 to 200 ppm introduces very little change in our conductivity estimates over assuming a 1:2:3 water partitioning, especially at the water contents of interest, around 80 ppm.
[53] The water content of minerals from xenoliths beneath Gibeon is unknown, but it is reasonable to assume that it will be of the same order, if not somewhat less. Undertaking the computations for Gibeon parameters (Table 3) yields the solid lines in Figure 6b for the Karato, Yoshino and Poe models.