The mean age of mantle and crustal reservoirs
Authors: Jacobsen Stein B. , Wasserburg G. J. 
Published: December 2012

Mantle and crust evolution is discussed in terms of two simple transport models. In model I, continents (j = 3) are derived by melt extraction over the history of the earth from undepleted mantle (j = 1), and the residue forms a depleted mantle (j = 2), which today is the source of mid‐ocean ridge basalts. In model II, new additions to continents are derived from a mantle reservoir 2, which becomes more depleted through time by repeated extraction of melts. Transport equations were solved for stable s, radioactive r, and daughter d isotopes for arbitrary mass growth curves Mj(τ). For both models the isotopic composition and concentrations of trace elements are shown to reduce to simple mathematical expressions which readily permit calculations of basic evolutionary parameters from the data. For longlived isotopes (λ−1 ≫ 4.5 aeons) for model I the deviations in parts in 104 of the ratio of a daughter isotope to a stable reference isotope of the same element in reservoirs j = 2, 3 from that of 1 is given by
. Here
is the mean age of the mass of
is the enrichment factor of the ratio of a radioactive isotope to a stable isotope relative to that in 1, and Qd* ≃ const. Thus for long‐lived isotopes such as 147Sm and 87Rb the only time information that can be obtained from model I from measurement of the relative chemical enrichment factors and isotopic ratios at a single time is the mean age of the mass of the continental crust and the complementary depleted mantle reservoir. This mean age is independent of the long‐lived parent‐daughter system. An analogous result is obtained for model II, where
. Here
is the weighted time average of the enrichment factor, and τ is the time measured from the origin of the earth. The mean age of the mass of the crust
and the time parameter
for the crust in model II will for long‐lived parent‐daughter systems be different depending on the element fractionation during partial melting. Decay systems with small parent‐daughter fractionations during partial melting may, however, be used to estimate the mean age of the continental crust. Sm‐Nd and Rb‐Sr isotopic data for continental crust, depleted and undepleted mantle, have been used to evaluate both models and yield young mean ages for the mass of the continental crust of 1.8 and 1.5 aeons for model I and model II, respectively. Both models also suggest that the rate of growth of the continents for the last 0.5 aeon is much less than the average growth rate. The young mean age of the continents implies either rapid refluxing of crustal materials to the mantle in the period from 4.5 to 3.6 aeons or that very little early crust ever formed. Mass balance calculations for both models show that the continents were only formed from ∼30% of the total mantle, leaving 70% of the mantle as undepleted. The major difference in the two models lies in the difference in the compositions of newly derived crust. For model I the trace element concentrations in new additions to the crust is constant, and the isotopic values are those of the undepleted mantle reservoir, in agreement with recent Nd isotopic studies. The correlation line between εNd and εSr for young basalts can be explained with model I by mixing depleted and undepleted mantle, but it cannot in any simple way be explained by model II. Model II implies that new additions to the continents have the isotopic characteristics of depleted mantle and that the concentration of Rb, U, Ba, and other highly incompatible trace elements in newly added material have changed by a factor of ∼10 through time, for which there is no evidence. For both models the simple analytical expressions derived herein permit calculations of earth models with great facility without requiring a computer calculation.