2.3.2. Constraints on Paleogene Mg/Casw: Coupled δ18O-Mg/Ca Data
 Some previous studies [e.g., Broecker and Yu, 2011; Creech et al., 2010; Lear et al., 2000, 2002] have argued that higher Cenozoic Mg/Casw values are more realistic, specifically in some cases that the model of Wilkinson and Algeo  is more likely to be correct than all other model and proxy data arguing for lower values, particularly in the Paleogene. As discussed above, this conclusion results from an incorrect assumption regarding the value of H in a Mg/Catest–Mg/Casw calibration.
 In principle, coupled δ18O-Mg/Ca measurements from foraminifera allow both temperature and to be evaluated: an approach taken by many previous studies to calculate changes in global ice volume over both the Cenozoic and the Quaternary [Elderfield and Ganssen, 2000; Lear et al., 2000; Billups and Schrag, 2002, 2003]. It is now apparent that there are in fact six variables to be constrained: , , temperature and the value of the constant H, as well as δ18Otest and Mg/Catestwhich can be analyzed in well-preserved samples. In an ice-free world, the assumption that deep waterδ18O was −1.2 or −0.9% (Lear et al.  and Cramer et al. , respectively), allows either or H to be calculated if the other is known. Because H is known for only a few species of foraminifera, while almost all of the proxy and model Mg/Casw evidence for the Paleogene suggests low (<2.5 mol mol−1) values (Figure 5), we show how the value of H can be calculated for a species where a calibration is not yet available, thereby fully reconciling Cenozoic Mg/Casw values lower than that of Wilkinson and Algeo  with δ18O-derived temperatures from foraminifera.
 Oridorsalis umbonatusis an extant benthic foraminifera which is also present throughout the Cenozoic, enabling direct comparison of recent and fossil material without the need for species-specific corrections. There are several Mg/Ca-temperature calibrations available for this species, which are not all in agreement. It is also unclear whether a linear or exponential fit more appropriately describes the data.Lear et al.  used an exponential calibration, whereas the data of Lear et al.  and Bryan and Marchitto  suggest a linear fit is more appropriate. In particular, calibrations focusing on low temperatures show that a linear fit is equally or more suitable. A linear or exponential best fit appear to represent the data of Rathmann et al.  equally well.
 Lear et al.  suggest that early Paleogene Mg/Casw values are likely to be in the region suggested by Wilkinson and Algeo , around 3.6 mol mol−1. This is because when H is assumed to be 1, lower Mg/Casw values shift the calibration to much higher temperatures for a given Mg/Catest value (or rather Mg/Catest values are shifted downward for a given temperature) and therefore produce reconstructed paleotemperature estimates in very poor agreement with comparative δ18O-derived results.
 At 49 Ma the Earth is assumed to be ice free, which, along with the aforementioned assumption of ocean bottom water δ18O on an ice-free planet and depending on whichδ18O-temperature calibration is used, has led to a range in reported temperatures for this time of 12.4–13.4°C [Lear et al., 2002; Cramer et al., 2011]. An uncertainty in the value of δ18Osw of 0.1% results in a temperature error of ∼0.4°C. By using the measured δ18O and Mg/Ca ratio of O. umbonatus at 49 Ma (2.9 mmol mol−1), the constant H can be calculated by combining both these data and respective temperature calibrations; at one specific value the Mg/Ca temperature will match precisely the δ18O-derived temperature. Because the model ofStanley and Hardie  is within error of the majority of the Mg/Casw data currently available (Figure 5), and an independent temperature estimate is available from δ18O measurement, the value of H for this species can be calculated. Specifically, we derive H by iteratively solving equation (6) so that T = 13.4°C when Mg/Casw = 1.6 mol mol−1 (the value given by Stanley and Hardie  at 49 Ma). This gives a value of Hfor this foraminifera species of 0.52 or 0.54 based on the Mg/Ca-temperature data ofRathmann et al.  and Lear et al. , respectively (Figure 6). This value of H is for δ18Osw = − 0.9%, a value of −1.2% would result in H = 0.44 or 0.48, respectively. Following Cramer et al.  a correction has been applied to those lines in Figure 6relating to paleo-Mg/Casw values for changes in the calcite compensation depth (CCD), using a CCD depth at 49 Ma of 3.25 km [Van Andel, 1975]. Such a correction is necessary as a relatively higher CCD lowers ΔCO32− for a given depth.
Figure 6. Reconciling the low proxy and model estimates of Paleogene Mg/Casw with data from foraminifera. Three calibrations that have been applied to fossil O. umbonatus are shown, that of Lear et al. [2002, 2010] and the data of Rathmann et al.  with the linear fit applied by Cramer et al. . Because δ18O in an ice-free world can be calculated andO. umbonatus Mg/Ca can be measured at a time when the world was ice free (49 Ma), this information can be used to calculate the value of H in this species, by solving equation (7). Doing so results in a range in H from 0.52 to 0.54 depending on which calibration is used (Lear et al.  and Cramer et al. , respectively), assuming a Paleogene Mg/Casw value of 1.6 mol mol−1 [Stanley and Hardie, 1998]. In the case of the calibration of Lear et al. , exactly the same result can be produced by using the far higher Mg/Casw value of Wilkinson and Algeo  at this time with the assumption that H = 1. We therefore show how two previously held incorrect assumptions (namely, that Paleogene Mg/Casw was <3 mol mol−1 and that the value of H= 1) can result in a sensible Mg/Ca-temperature estimate, leading to erroneous foraminiferal constraints on Mg/Casw. It is not possible to recreate the δ18O-derived temperature using the calibration ofCramer et al.  coupled to the Mg/Casw of Wilkinson and Algeo , adding support to this argument. The calibration of Lear et al. , with a significantly lower slope, would appear to suggest H ≈ 0.
Download figure to PowerPoint
 In the absence of any calibration study this result should be treated as preliminary, particularly as Mg/Casw is not very well constrained throughout much of the Cenozoic. In section 2.3.1 we noted that with one exception all of the Mg/Casw proxy evidence lies within ±0.5 mol mol−1 of the model of Stanley and Hardie . However, propagating this error through to our calculation of Hresults in an uncertainty of ∼±0.14, therefore until early mid-Paleogene Mg/Casw is better constrained, this methodology can be used as a guide only. Furthermore, an uncertainty in of ±0.1% results in an error in H of approximately ±0.04.
 Despite considerable uncertainties in the calculation of H, we demonstrate how it is possible to reconcile low Mg/Casw values with δ18O-derived temperatures from foraminifera. Given that for all species studied so farH < 1, the possibility of high (>2.5 mol mol−1) Paleogene Mg/Casw values are precluded. Previously, constraints have been placed on Paleogene Mg/Casw by assuming H = 1. This resulted in the exclusion of estimates from the lower range of Mg/Casw values, as a Paleogene ratio of 3.6 mol mol−1 was required to match the δ18O with the Mg/Ca-derived temperatures, despite the available proxy evidence suggesting that such higher values were unlikely. The reasons for this result are discussed above, however it should again be noted that this aspect of Mg/Ca paleothermometry was only understood after the publication of the majority of studies that assumeH = 1. Because virtually all of the paleo-Mg/Casw direct proxy evidence lies reasonably close to the model of Stanley and Hardie  and H < 1 for all foraminifera species studied so far, it is not possible that the model of Wilkinson and Algeo  is representative of the magnitude of increase in Cenozoic Mg/Casw.
 Recently, Cramer et al.  calculated a Cenozoic Mg/Casw record by combining the Cenozoic benthic foraminifera Mg/Ca and δ18O records with a sea level record (used as a proxy for ice volume). We outline in this contribution the inaccuracy in previous Mg/Casw corrections and the associated assumptions and constraints placed on Paleogene Mg/Casw. The recent reconstruction of Cramer et al.  is therefore not assessed in detail here, as such a discussion is outside the intended remit of this paper. Briefly, however, the Cramer et al.  Mg/Casw reconstruction is not included in Figure 5 because it is not entirely independently derived. The record was produced by matching the Cenozoic Mg/Ca and δ18O temperature curves by varying Mg/Casw at each time interval and therefore requires some assumption of the value of H or Mg/Casw at one or several tie points. Given that H is not known from culturing for any deep benthic foraminifera species, the assumption is of Mg/Casw and it is therefore not independent of the reconstructions shown in Figure 5. This is important because the relatively high Mg/Casw values that Cramer et al.  reconstruct (broadly >3 mol mol−1) are a result of their assumption regarding Mg/Casw at some point in the past. Assuming a lower Mg/Casw value at this time would shift the entire reconstruction downward and it therefore does not provide independent evidence that lower Mg/Casw estimates are incorrect.
 Although the discussion has so far focused on the Paleogene, because seawater Mg/Ca ratios were significantly different before ∼20 Ma in comparison to the present day, it is also important to appropriately adjust Mg/Ca data from fossil material younger than this. The model of Fantle and DePaolo  (Figure 5), based directly on high-resolution proxy evidence, suggests that the Mg/Ca ratio of seawater may have undergone fluctuation on submillion year timescales significant enough to greatly increase the inaccuracy in Mg/Ca-derived temperatures. If Mg/Caswhas indeed undergone far more short-term fluctuation than previously accounted for, then any study based on samples older than 0.5 Ma reporting absolute temperatures should take the correction outlined here into account [seeMedina-Elizalde et al., 2008]. Furthermore, any high-resolution study spanning more than 0.5–1 Ma should consider the effect of potential changes in Mg/Casw. While the majority of the Mg/Casw proxy data agree well with the model of Stanley and Hardie , further high-resolution model and proxy reconstructions of secular variation in Cenozoic Mg/Casw are clearly a priority.