I’m ostensibly working on a project which involves doing some bookkeeping on where rare-Earth elements go during magma ocean solidification of Earth’s early mantle.  After happening in fits and spurts for longer than I’d care to admit, there was a bit of a breakthrough tonight, thanks to the efforts of my brilliant chemical engineer housemate.

Our model splits Earth’s mantle into 1000 concentric shells, then solidifies the shells layer by layer from the bottom up (because the adiabatic temperature profile and the solidus intersect at the bottom of the mantle).  Assuming that once a layer has solidified, the rest of the mantle evenly mixes to have the same composition means you can just think about this problem two layers at a time: the layer you’re solidifying (with pre-solidification composition measured in mass percentage liquid_n and mass mass_n), and the layer just above that.  The above layer will have the same composition as the rest of the mantle (liquid_n+1), which means you can just concentrate (ha) on those two layers without having to worry about the mass of the rest of the mantle.

We’re also keeping track of the composition (solid) and mass (mass (solid)_n) of what fractionates (solidifies) out of the liquid, as well as the volumes of each shells (vol).

With all these parameters in mind, we can calculate the composition of the next layer of liquid (and by proxy, the rest of the mantle) after solidification liquid_n+1:

This equation assumes that mixing of the remaining liquids post-solidification in the magma ocean occurs on small timescales compared to that of the solidification process, producing a homogenous liquid mantle.  The magma ocean is assumed to have a bulk silicate mantle of melted material with a composition from Hart and Zindler (1986) and Bertka and Fei (1997), with average chondritic trace elements from Anders and Grevesse (1989). For solidified minerals, mineral phase behavior is based on experimental results (Elkins-Tanton et al., 2003; Trønnes and Frost, 2002; Bertka and Fei, 1997).

Why bother with going to all this trouble for mass balance?  The previous equation for calculating the composition of the next liquid layer (and thus the remainder of the mantle’s magma ocean) would make some of the mass percentages in that layer go to zero, or worse, negative, meaning that the rest of your model was being fed rather bad values.  Now, after much digging and wrangling through MATLAB’s debugging mode (not to mention initially blaming some nuances of clinopyroxene and magnesiowüstite’s density behavior at various temperatures and pressures ), it looks like the problem might have been in how the composition of the next liquid layer was calculated, or at least the problem has shifted to other points in the model.  Although the liquid composition is no longer getting assigned negative values, there’s no certainty that the values being written into it are reasonable, much less physically sane.

Regardless: thank you, generous and insightful housemate!  Next on deck: what common mantle rock-forming minerals have densities greater than 4,000 kg m^-3?  What happens to the last 0.03% of the mantle that doesn’t solidify?  What’s the viscosity and density of mantle material below the magma ocean and above the core?  How do you explain that melting starts where the adiabat intersects the solidus?  Stick with us next time for As The Earth Solidifies (And Overturns)!