What can we learn from REE abundances in clinopyroxene and orthopyroxene in residual mantle peridotites?

Abstract

Clinopyroxene and orthopyroxene are the two major repositories of rare-earth elements (REE) in spinel peridotites. Most geochemical studies of REE in mantle samples focus on clinopyroxene. Recent advances in in situ trace element analysis have made it possible to measure REE abundance in orthopyroxene. The purpose of this study is to determine what additional information one can learn about mantle processes from REE abundances in orthopyroxene coexisting with clinopyroxene in residual spinel peridotites. To address this question, we select a group of spinel peridotite xenoliths (9 samples) and a group of abyssal peridotites (12 samples) that are considered residues of mantle melting and that have major element and REE compositions in the two pyroxenes reported in the literature. We use a disequilibrium double-porosity melting model and the Markov chain Monte Carlo method to invert melting parameters from REE abundance in the bulk sample. We then use a subsolidus reequilibration model to calculate REE redistribution between clinopyroxene and orthopyroxene at the extent of melting inferred from the bulk REE data and at the closure temperature of REE in the two pyroxenes. We compare the calculated results with those observed in clinopyroxene and orthopyroxene in the selected peridotitic samples. Results from our two-step melting followed by subsolidus reequilibration modeling show that it is more reliable to deduce melting parameters from REE abundance in the bulk peridotite than in clinopyroxene. We do not recommend the use of REE in clinopyroxene alone to infer the degree of melting experienced by the mantle xenolith. In general, HREE in clinopyroxene and LREE in orthopyroxene are more susceptible to subsolidus redistribution. The extent of redistribution depends on the modes of clinopyroxene and orthopyroxene in the sample and thermal history experienced by the peridotite. By modeling subsolidus redistribution of REE between orthopyroxene and clinopyroxene after melting, we show that it is possible to discriminate mineral mode of the starting mantle and cooling rate experienced by the peridotitic sample. We conclude that endmembers of the depleted MORB mantle and the primitive mantle are not homogeneous in mineral mode. A modally heterogeneous peridotitic starting mantle provides a simple explanation for the large variations of mineral mode observed in mantle xenoliths and abyssal peridotites. Finally, using different starting mantle compositions in our simulations, we show that composition of the primitive mantle is more suitable for modeling REE depletion in cratonic mantle xenoliths than the composition of the depleted MORB mantle.

Appendix: Governing equations and inversion method

The disequilibrium double-porosity melting model of Liang and Liu (2016) is intended for modeling trace element fractionation during concurrent melting, channelized melting extraction, and finite rate of crystal-melt chemical exchange in an upwelling steady-state melting column. In terms of the degree of melting experienced by the solid matrix (F), mass conservation equations for a trace element in the interstitial melt ($C_{{\text{f}}}$), residual solid ($C_{{\text{s}}}$), individual mineral ($C_{{\text{s}}}^{j}$) can be written as:

$$\varepsilon_{{{\text{cpx}}}} \left( {1 - {\mathbb{R}}} \right)F\frac{{{\text{d}}C_{{\text{f}}} }}{{{\text{d}}F}} = \varepsilon_{{{\text{cpx}}}} \left( {C_{{\text{s}}}^{p} - C_{{\text{f}}} } \right) + \mathop \sum \limits_{j = 1}^{N} w_{j} \frac{{R_{j} }}{{R_{{{\text{cpx}}}} }}\left( {C_{{\text{s}}}^{j} - k_{j} C_{{\text{f}}} } \right),$$

(8)

$$\varepsilon_{{{\text{cpx}}}} { }\left( {1 - F} \right)\frac{{{\text{d}}C_{{\text{s}}} }}{{{\text{d}}F}} = \varepsilon_{{{\text{cpx}}}} \left( {C_{{\text{s}}} - C_{{\text{s}}}^{p} } \right) - \mathop \sum \limits_{j = 1}^{N} w_{j} \frac{{R_{j} }}{{R_{{{\text{cpx}}}} }}\left( {C_{{\text{s}}}^{j} - k_{j} C_{{\text{f}}} } \right)$$

(9)

$$\varepsilon_{{{\text{cpx}}}} { }\left( {1 - F} \right)\frac{{{\text{d}}C_{{\text{s}}}^{j} }}{{{\text{d}}F}} = - \frac{{R_{j} }}{{R_{{{\text{cpx}}}} }}\left( {C_{{\text{s}}}^{j} - k_{j} C_{{\text{f}}} } \right).$$

(10)

where $k_{j}$ is the partition coefficient between mineral j and melt for the element of interest; ${\mathbb{R}}$ is the dimensionless melt suction rate defined as the fraction of melt removed from the residual solid (to a nearby channel) relative to the amount of melt produced by melting (Iwamori 1994; Liang and Peng 2010); $\varepsilon_{{{\text{cpx}}}}$ is the disequilibrium parameter for a trace element in cpx; $C_{{\text{s}}}^{p}$ is the concentration of bulk solid calculated according to melting reaction; and $w_{j}$ is the weight fraction of mineral j in residual solid. The latter three parameters are defined as follows:

$$\varepsilon_{{{\text{cpx}}}} = \frac{\Gamma }{{\rho_{{\text{s}}} \left( {1 - \phi_{{\text{f}}} } \right)R_{{{\text{cpx}}}} }}$$

(11)

$$C_{{\text{s}}}^{p} = \mathop \sum \limits_{j = 1}^{N} w_{j}^{p} C_{{\text{s}}}^{j}$$

(12)

$$w_{j} = \frac{{w_{j}^{0} - w_{j}^{p} F}}{1 - F},$$

(13)

where $\Gamma$ is the melting rate; $\rho_{{\text{s}}}$ is the density of the bulk solid; $\phi_{{\text{f}}}$ is the volume fraction of the melt in the residue; $w_{j}^{0}$ is the weight fraction of mineral j at the onset of the melting; and $w_{j}^{p}$ is the weight fraction of mineral j participating in the melting reaction. Equations (8–10) can be further simplified by assuming that the interstitial melt and olivine and spinel are in local chemical equilibrium. The mass conservation equations become:

$$\begin{aligned} \left( {\varepsilon_{{{\text{cpx}}}} \left( {1 - {\mathbb{R}}} \right)F + \varepsilon_{{{\text{cpx}}}} \left( {1 - F} \right)\left( {w_{{{\text{ol}}}} k_{{{\text{ol}}}} + w_{{{\text{sp}}}} k_{{{\text{sp}}}} } \right)} \right)\frac{{{\text{d}}C_{{\text{f}}} }}{{{\text{d}}F}} & = \varepsilon_{{{\text{cpx}}}} \left( {C_{{\text{s}}}^{p} - C_{{\text{f}}} } \right) \\ & \quad + w_{{{\text{cpx}}}} \left( {C_{{\text{s}}}^{{{\text{cpx}}}} - k_{{{\text{cpx}}}} C_{{\text{f}}} } \right) + w_{{{\text{opx}}}} \frac{{R_{{{\text{opx}}}} }}{{R_{{{\text{cpx}}}} }}\left( {C_{{\text{s}}}^{{{\text{opx}}}} - k_{{{\text{opx}}}} C_{{\text{f}}} } \right) \\ \end{aligned}$$

(14)

$$\varepsilon_{{{\text{cpx}}}} \left( {1 - F} \right)\frac{{{\text{d}}C_{{\text{s}}}^{{{\text{cpx}}}} }}{{{\text{d}}F}} = - \left( {C_{{\text{s}}}^{{{\text{cpx}}}} - k_{{{\text{cpx}}}} C_{{\text{f}}} } \right)$$

(15)

$$\varepsilon_{{{\text{cpx}}}} \left( {1 - F} \right)\frac{{{\text{d}}C_{{\text{s}}}^{{{\text{opx}}}} }}{{{\text{d}}F}} = - \frac{{R_{{{\text{opx}}}} }}{{R_{{{\text{cpx}}}} }}\left( {C_{{\text{s}}}^{{{\text{opx}}}} - k_{{{\text{opx}}}} C_{{\text{f}}} } \right)$$

(16)

$$C_{{\text{s}}}^{p} = w_{{{\text{cpx}}}}^{p} C_{{\text{s}}}^{{{\text{cpx}}}} + w_{{{\text{opx}}}}^{p} C_{{\text{s}}}^{{{\text{opx}}}} + w_{{{\text{ol}}}}^{p} k_{{{\text{ol}}}} C_{{\text{f}}} + w_{{{\text{sp}}}}^{p} k_{{{\text{sp}}}} C_{{\text{f}}} .$$

(17)

Concentration of REE in the bulk residue can be calculated using the expression:

$$C_{{\text{s}}} = \sum w_{j} C_{{\text{s}}}^{j} = w_{{{\text{cpx}}}} C_{{\text{s}}}^{{{\text{cpx}}}} + w_{{{\text{opx}}}} C_{{\text{s}}}^{{{\text{opx}}}} + w_{{{\text{ol}}}} k_{{{\text{ol}}}} C_{{\text{f}}} + w_{{{\text{sp}}}} k_{{{\text{sp}}}} C_{{\text{f}}} .$$

(18)

Equations (14–16) are closed by the following boundary conditions at the bottom of the melting column (F = 0):

$$C_{{\text{f}}} \left( 0 \right) = \frac{{\varepsilon_{{{\text{cpx}}}} { }k_{p} + k_{0} }}{{k_{0} \left( {\varepsilon_{{{\text{cpx}}}} + k_{0} } \right)}}C_{{\text{s}}}^{0}$$

(19)

$$C_{{\text{s}}}^{{{\text{cpx}}}} (0) = \frac{{k_{{{\text{cpx}}}} }}{{k_{0} }} C_{{\text{s}}}^{0}$$

(20)

$$C_{{\text{s}}}^{{{\text{opx}}}} \left( 0 \right) = \frac{{k_{{{\text{opx}}}} }}{{k_{0} }}C_{{\text{s}}}^{0} .$$

(21)

where $k_{0}$ is the bulk solid–melt partition coefficient at the onset of melting; and $k_{p}$ is the bulk solid–melt partition coefficient according to the melting reaction. Equation (19) is based on the analysis of Liang and Liu (2016). Equations (14–16) are a set of coupled ordinary differential equations and can be solved numerically using standard methods.

The disequilibrium double-porosity melting model consists of three coupled nonlinear ordinary differential equations and one algebraic equation for spatial variations of a trace element in interstitial melt, cpx, opx, and bulk residue (Eqs. 14–17). They have no analytical solutions for the problem considered in this study. Hence, we cannot use nonlinear least squares analysis to invert melting parameters as we did for REE in cpx (Liang and Peng 2010; Liang and Liu 2016). To get around this difficulty, we use Markov chain Monte Carlo (MCMC) method to invert the melting parameters F, $\varepsilon_{{{\text{cpx}}}}^{1300}$, and ${\mathbb{R}}$ from REE + Y concentrations in the bulk sample. MCMC methods are a class of powerful statistical tools for solving nonlinear inverse problems in Bayesian analysis (e.g., Sambridge and Mosegaard 2002). Liu and Liang (2017) used MCMC to invert melting parameters from REE abundance in residual cpx in abyssal peridotites. The present work follows the inversion procedure detailed in their study. To facilitate MCMC inversion, we set bounds for the disequilibrium parameter $\varepsilon_{{{\text{cpx}}}}^{1300}$ to [0, 0.15] and the melt suction rate ${\mathbb{R}}$ to [0, 1], based on previous inversion results using similar model or method (Lundstrom 2000; Liang and Peng 2010; Liang and Liu 2016; Liu and Liang 2017). We solve Eqs. (14–21) for each simulation using the routine ODEIENT in Python. We run 10,000–50,000 simulations for each sample to ensure convergence and use the result to calculate best fitting parameters and uncertainties. The best fitting parameters are then used to calculate REE abundances in the bulk sample, residual cpx and opx, and chemically re-equilibrated cpx and opx at closure temperatures in the two pyroxenes.