Kappa conundrum--The relationship between measured kappa ratio and time-integrated kappa ratio of the Earth’s depleted mantle

1.  Background

The ratio of ²³²Th/²³⁸U is refered to as kappa (κ), and is calculated by two methods. First, knowing the abundance of the isotopes of Th and U yields a measured kappa ratio (κ_MEAS). Second, the lead isotope ratio (²⁰⁸Pb*/²⁰⁶Pb*, “*” means this ratio is the lead radiogenic ratio) yields a time-integrated kappa ratio (κ_Pb), because ²⁰⁸Pb and ²⁰⁶Pb are the decay products of ²³²Th and ²³⁸U respectively [2]. Considering how the two kappa ratios are calculated, κ_MEAS and κ_Pb should equal to each other in a certain reservoir. However, it is known that κ_MEAS and κ_Pb are different from each other in the continental crust as well as in the mantle. To be more specific, the measured κ _MEAS(C) is higher than the κ_Pb(C) of continental crust, while the κ _MEAS(M) is lower than the κ_Pb(M) of mantle. Such strange phenomenon is commonly called the kappa conundrum. [3]

2. The goal of study

3. Current results

4. Possible mechanism for the kappa conundrum

The recent melt fractionation of Th and U seems to be a reasonable explanation for the kappa conundrum. U series disequilibrium (²³⁸U -²³⁰Th) provides evidence of the recent fractionation of Th and U [4,5]. However, both Th and U are highly incompatible elements and have similar partition coefficients, and the evidence that prevalent ²³⁰Th excesses (about 20%) in MORB suggests that melting during magma genesis does not cause enrichment of U compare to Th in MORB [2]. Thus, there must be other explanation(s) to the kappa conundrum.

A Monte Carlo simulation to investigate the possible pathways of κ evolution. All the simulations are started at the bulk silicate earth kappa ratio (κ_BSE), which equals to 4.0±0.2. Each simulation is developed within 80 equal steps of 0.05688Ga, and within each steps, the κ values randomly changed between -0.15 to +0.15. The successful pathways are the ones that result in present values of κ_MEAS and κ_Pb of 2.5-2.7 or 3.7-3.8, respectively. I did the Monte Carlo Simulation for 1000 times for measured kappa ratio, with each κ starting at 4.0 and ending up within the range of 2.4 to 2.6 for measured kappa ratio.

Monte Carlo simulation of κ_Pb 

The Monte Carlo simulation result suggests that a major decline of κ value in the post-Archean period is required in the history of the kappa ratio development, which means the abundance of Th and/or U in the mantle and crust need to be changed.

As we have discussed above, the melting fractionation is not an option. But there is always more than one way to separate Th and U. A well-recognized mechanism that could account for the decline of κ value in the post-Archean period is the different solubility of U and Th’s ions under oxidized environments. [6]

Under natural conditions, Th has only one state of oxidation, which is the highly insoluble ion Th⁴⁺. On the other hand, U has more than one oxidation states, which are U⁴⁺, U⁵⁺ and U⁶⁺. And among those, U⁶⁺ behaves very differently from Th⁴⁺ and becomes a highly soluble ion. As a result, compared to Th, U is more easily to be carried away by the river fluids during weathering, and enter the ocean. Two possibilities will occur under this situation. First, all of the U will sink into the oceanic crust, and be directly returned to the continental crust at the subduction zones, which will keep the kappa ratio unchanged but is also unlikely to happen [7,8]. Second, at least some of the sank U will be recycled back to the mantle by the act of plate tectonic cycle, which will lead to the continuously decreasing of the mantle measured kappa ratio [6].

The change of solubility of U ions can only affect the measured kappa ratio of the Earth’s mantle and crust, but not the time-integrated kappa ratio. The reason of it is the lost and gaining of U in the crust and mantle respectively will directly affect the U abundance, which is the denominator of the measured kappa ratio equation . Thus, the measured kappa ratio of the crust will increase and the mantle’s will decrease comparing to the original values. However, both ²³⁸U and ²³²Th are long-lived isotopes, so the recent geological activities will not result in their decay daughter elements’ abundance change. With T and decay constants unchanged, the time-integrated kappa ratio is only proportional to the ²⁰⁸Pb*/²⁰⁶Pb* ratio. Thus, the lead isotope kappa ratio will not be affected. Since the measured kappa ratio changed while lead kappa ratio remains unchanged, it will certainly result in the unequal values of the two ratios, which is consistent with our calculation results.

5. Future plan
      

References

[1] Paul, D., W. M. White, and D. L. Turcotte. "Constraints on the 232Th/238U ratio (κ) of the continental crust." Geochemistry, Geophysics, Geosystems 4.12 (2003).

[2] Galer, S. J. G., and R. K. O'Nions. "Residence time of thorium, uranium and lead in the mantle with implications for mantle convection." (1985): 778-782.

[3] Elliott, Tim, Alan Zindler, and Bernard Bourdon. "Exploring the kappa conundrum: the role of recycling in the lead isotope evolution of the mantle." Earth and Planetary Science Letters 169.1 (1999): 129-145.

[4] B. Bourdon, A. Zindler, T. Elliott, C.H. Langmuir, Constraints on mantle melting at mid-ocean ridges from global 238U=230Th disequilibrium data, Nature 384 (1996) 231– 235.

[5] C.C. Lundstrom, Q. Williams, J.B. Gill, Investigating solid upwelling rates beneath mid-ocean ridges using U-series disequilibria, 1: a global approach, Earth Planet. Sci. Lett. 157 (1998) 151–165.

[6] Elliott, Tim, Alan Zindler, and Bernard Bourdon. "Exploring the kappa conundrum: the role of recycling in the lead isotope evolution of the mantle." Earth and Planetary Science Letters 169.1 (1999): 129-145.

[7] Gill, James B., and Ross W. Williams. "Th isotope and U-series studies of subduction-related volcanic rocks." Geochimica et Cosmochimica Acta 54.5 (1990): 1427-1442.

[8] McDermott, Frank, and Chris Hawkesworth. "Th, Pb, and Sr isotope variations in young island arc volcanics and oceanic sediments." Earth and Planetary Science Letters 104.1 (1991): 1-15.