A theoretical approach to understanding the isotopic heterogeneity of mid-ocean ridge basalt

We have developed an idealized mathematical model to understand the isotopic variability of the mantle and its relation to the observed variations in isotopic ratios ¹⁴³Nd/¹⁴⁴Nd, ⁸⁷Sr/⁸⁶Sr, ¹⁷⁶Hf/¹⁷⁷Hf, ²⁰⁸Pb/²⁰⁴Pb, ²⁰⁶Pb/²⁰⁴Pb, and ²⁰⁷Pb/²⁰⁴Pb measured on mid-ocean ridge basalt (MORB). We consider a simple box model of mantle processes. A single melt region produces a melt fraction F of melt, and the average time since a given parcel of mantle material last visited this region is given by the time scale τ_melt. The melt region fractionates the parent/daughter ratios. Over time this leads to variations in the mantle isotopic ratios as the parent decays to the daughter. Key assumptions are that the half-life of the parent isotope is large compared with τ_melt, that the flow is strongly stirring, and that the mantle has reached a statistical steady state. This enables us to neglect the specifics of the underlying flow. Sampling from our model mantle is dealt with by averaging over a large number N of samples to represent the mixing after melting.The model predicts a probability density for isotopic ratios in MORB which, with exception of the Pb isotopes, are consistent with measurements. Fitting the MORB data to this model gives estimates of the model parameters F, τ_melt, and N. Small melt fractions with F around 0.5% are essential for a good fit, whereas τ_melt and N are less well constrained. τ_melt is estimated at around 1.4 to 2.4 Ga, and N is of the order of hundreds. The model predicts a larger variability for the Pb isotopes than that observed. As has been stated by many previous authors, it appears that fundamental differences exist between the dynamics of Pb isotopes and those of Nd, Sr and Hf isotopes.