D R. Stegman $^{a}$, H. M. Gonnermann $^{a}$, M. Manga $^{a}$, and M. A. Richards $^{a}$
$^{a}$ University of California, Berkeley, California
We extend parameterized, layered convection models (e.g. McNamara and Van Keken, 2000; Spohn and Schubert, 1982) to include for heat and mass exchange due to entrainment of material across the chemical density interface (CDI) of two fluid layers. The rate of entrainment, $Q$ (measured as volumetric flux) is accounted for by a scaling relationship, confirmed with laboratory experiments of thermo-chemical convection (Davaille, 1999; Gonnermann et al, 2002) which found $Q$ to be inversely proportional to the square of the Buoyancy number, $B^{-2}$, and proportional to the Rayleigh number, $Ra^{1/3}$. We assume that the total amount of heat exchange across the CDI is composed of both a conductive component and an advective component due to entrainment. Furthermore, we assume that advective component is directly proportional to entrainment rate. Entrainment is the sole mechanism which enables a layered system to evolve towards a single layer with uniform composition (in both density and heat producing elements) and single temperature. Previous layered convection models (with 1600 km thick upper layer) assume that the heat transport across the CDI is purely by conduction. These models are required to satisfy a reduced mantle heat flow of 40 TW (total surface heat flow of 44 TW minus a contribution by radioactivity in the crust). The contribution from radioactivity in the continental crust in their models is assumed to be only 4 TW of the total, but this assumed value is probably underestimated and should actually be 8-10 TW. Satisfying a reduced mantle heat flow of 40 TW (and also 34 TW) under the purely conductive assumption commonly results in models which exceed an average lower layer temperature of 4000 K at some point during the model's 4.5 Gyr thermal history (McNamara and Van Keken, 2000). Since the internal heat production of the upper (depleted) layer accounts for only 10\% of the reduced mantle heat flow, the remaining amount must be conducted across the CDI (note that due to spherical geometry, surface area of the CDI is only ½ that of the surface). Thus, the thermal boundary layer which coincides with the CDI has a temperature difference of ($\approx$3000 K), and consequently results in supersolidus lower layer temperatures. We reproduce results reported by McNamara and Van Keken (2000) for the case of layered convection in which all heat must be conducted across the CDI. These models further assume symmetrical thermal boundary layers in the upper layer, a critical Ra equal to 1100, and constant viscosity. For this model, none of the 18 calculated thermal histories were able to meet all three constraints: (1) providing 40 TW of surface heat flow, (2) having a final upper layer temperature greater than 1500 K, and (3) a lower layer temperature never exceeding 4000 K. We then account for heat transport by entrainment, while keeping the density difference constant and allowing the heat production of each layer to evolve independently, and find that 4 out of 18 models meet all constraints. These successful models illustrate the significant effect that entrainment has on thermal evolution by introducing a non-conductive heat transfer across the CDI, which consequently keeps the temperature difference across the CDI remains small ($\approx$300 K). An additional, yet much more uncertain, constraint is provided by bulk silicate earth (BSE) models which propose a Uranium concentration of 21 ppb in the undifferentiated mantle (Van Schmus, 1995). This amount of radioactivity can account for 20 TW of the observed 44 TW surface heat flow (leading to a Urey number of approximately 0.5). Furthermore, assuming that $\approx$30-50\% of the heat producing elements [U, Th, and K] were differentiated into the continental crust and that the present day Uranium concentration of depleted upper mantle is 7 ppb, we can estimate the remaining amount of heat producing elements in an assumed 1300 km thick lower layer as 18-26 ppb. In addition to including for advective heat transport, we also incorporate mass transfer across the CDI in our parameterized thermo-chemical model. For models beginning with a 5\% density stratification and evolving towards a uniform composition, we are unable to simultaneously satisfy the surface heat flow, mantle temperatures, and lower layer Uranium concentrations provided by BSE models. However, it is clear that initial conditions play a more important role in determining these thermal histories, and a number of factors (initial density contrast, initial temperature of each layer, initial Uranium concentration for each layer, viscosity of each layer, etc.) may trade off against each other. We find that initial density contrasts of $>$ 5\% (evolving towards a present day density contrast of 1-2\% and \$approx$300-400 K temperature difference) may have some combination of initial conditions that may satisfy heat flow and BSE constraints. Similarly, different lower layer thicknesses may also possibly satisfy such constraints. We conclude that including the heat and mass transfer due to entrainment results in more reasonable thermal histories for layered models, but it is unlikely a lower mantle layer of any size (stealth or otherwise) can avoid evolving towards a uniform composition with the upper layer. Therefore, we hypothesize that replenishment of the lower layer in enriched (but not necessarily more dense) material may play a significant role in maintaining a large difference in Uranium concentration between layers in the face of mass exchange due to entrainment. However, the amount of partitioning of radioactive elements between layers is entirely a construct of current BSE models providing a Urey number of 0.5. Given that the uncertainty in BSE models is much larger than that of present day observed surface heat flux, the Urey number (and corresponding difference in Uranium concentration between layers) should be viewed with caution when applied as a constraint on mantle composition and structure. Future work will investigate the possible role of sequestering radioactive elements in oceanic crust and replenishing a lower mantle geochemical reservoir. References Davaille, A., Simultaneous generation of hotspots and superswells by convection in a heterogeneous planetary mantle, {\it Nature, 402,} 756-760, 1999. Gonnermann, H. M., M. Manga and A. M. Jellinek, Dynamics and longevity of an initially stratified mantle, {\it Geophys. Res. Lett., 29,} #10.1029/2002GL01485, 2002. McNamara, A. K., and P. E. van Keken, Cooling of the Earth: A parameterized convection study of whole versus layered models, {\it G$^3$, 1,} #2000GC000045, 2000. Spohn, T., and G. Schubert, Modes of mantle convection and the removal of heat from the Earth's interior, {\it J. Geophys. Res., 87,} 4682-4696, 1982. Van Schmus, W. R., Natural Radioactivity of the crust and mantle, in {\it Global Earth Physics, A Handbook of Physical Constants, AGU Ref. Shelf,} vol. 1, edited by T.J. Ahrens, 283-291, AGU, Washington, D.C., 1995.