Mixing in numerical models of mantle convection incorporating plate kinematics

The process by which subducted lithosphere is mixed by mantle convection is investigated in numerical calculations. The results show that the observed isotopic heterogeneity of mantle sources and their ancient (1-2 b.y.) apparent ages are consistent with convective mixing. Passive tracers, which are introduced below "trenches," are efficiently dispersed, but nonetheless, heterogeneities in tracer density with a large range of length scales are observed to persist for 40 or more transit times (one transit time is the time to travel the fluid depth with the boundary velocity). In particular, there is a strong tendency to form high-density folds of the tracer strings, which persist much longer than simple shearing indicates. The folds persist because there is a strong tendency for material that enters the flow at the margins of cells to be transferred to adjacent cells, where it is "unmixed." When the simulations are scaled to the whole mantle, the tight clumps (folds) of tracers are shown to persist for up to 1-2 b .y. There is also a tendency for large-scale convection cells to remain isolated from recycled material for 1-2 b.y. These results are consistent with the significant chemical heterogeneity of the mantle as revealed by isotopic studies of oceanic basalts. Despite the spatial heterogeneity in tracer density, the average time tracers remain in the box from subduction at trenches to sampling at ridges (i.e., the residence time) is well constrained and within 20% of the mean residence time expected from an analytic model in which tracers are assumed to be sampled randomly. Model ages of the mantle that explicitly incorporate increased convection rates in the past and assume random sampling of heterogeneities bracket the - 2 b.y. apparent Pb-Pb and Rb-Sr isochrons of midocean ridge basalts and oceanic island basalts. The conclusion of persistent spatial heterogeneity is different from the conclusions drawn from other studies. The different conclusions result, primarily, from our emphasis on the details of spatial variations as opposed to some average of the mixing, from a difference in flow unsteadiness, and from the different ways tracers have been introduced into the flow

Numerical study of high Rayleigh number convection in a medium with depth-dependent viscosity

The equations of motion are solved numerically for a Boussinesq fluid with infinite Prandtl number in a square 2-D box where the viscosity increases with depth. Three heating modes are employed: bottom heating, internal heating, and half bottom and half internal heating. In all cases the boundaries are free slip. The range of Rayleigh numbers employed is 10^4-10^7. The viscosity increases as 10^(β(1-y)), where y is distance measured from the bottom upwards and β is a free parameter. In the bottom heated cases, the convective velocities slow near the bottom and result in a large temperature drop between the bottom boundary and interior compared with the top boundary and the interior. This results in increased buoyancy in the ascending limb. In the internally heated case, the flow in the top half of the box resembles Rayleigh-Bènard convection and in the bottom half it approaches a conductive thermal regime for β greater than about 2. In this case the top surface heat flux decays from ascending to descending limb and the ascending and descending limbs become more equal in their buoyancy. Increasing β decreases the efficiency of heat transport, but has little effect on the exponents of Nu-Ra and Pe-Ra relations. There is a larger decrease in heat transport efficiency for a given β in the bottom heated case compared to the internally heated case

Measurement of elastic velocities of MgO under shock compression to 500 kilobars

The velocities of rarefaction waves in shock-compressed MgO have been measured by observing the reduction of the shock front velocity near the sample edges due to the rarefaction waves propagating from the edges. The extent of this ‘edge effect’ is difficult to determine accurately because of its emergent nature. Arrangements sensitive to differences in shock front velocity yielded rarefaction wave velocities close to predicted longitudinal velocities in the high-pressure shock state. Velocities closer to the hydrodynamic sound speed in the shock state were obtained from less sensitive arrangements. These results can be interpreted in terms of a two-stage elastoplastic model of the decompression. The longitudinal velocities measured in shock states up to 528 kb imply second pressure derivatives of the elastic moduli c_(ij)″, given by K_0c_(ij)″ = −1 ± 15, where K is the bulk modulus

A proposed equation of state of stishovite

The available shock-wave data for solid α quartz in the stishovite pressure regime are reduced to a 25°C isotherm and an adiabat, centered at standard conditions, using recent standard density, enthalpy, and coefficient of thermal expansion data. The calculated isothermal bulk modulus, 3 Mb, as determined from the Birch-Murnaghan equation, depends critically on the value of (dK/dP)T at zero pressure and to a yet unknown extent on the form of the equation of state. The high-temperature value of Grüneisen's ratio (0.8 to 0.9) along the a quartz (stishovite regime) Hugoniot was obtained from the pressure offsets of the fused quartz and porous quartz Hugoniot. The high value for γ obtained from thermochemical data at standard conditions (1.5±0.3) suggests that a marked decrease in the value of γ to 0.8 occurs with increasing temperature

Math and Science Education

Resnick provides an excellent brief account of current work in cognitive psychology and its important implications for math and science education. As she indicates, most cognitive psychologists view knowledge as consisting of highly organized schemata into which new experiences are assimilated and view the learner as actively constructing new knowledge. This view is consistent with the ideas that Piagetian theorists and educators have been propounding for many years, although Resnick’s discussion is rooted in the more detailed analysis of specific knowledge and learning in specific content areas that typifies the information-processing paradigm of modern cognitive science

Interaction of mantle dregs with convection: Lateral heterogeneity at the core-mantle boundary

Preliminary numerical models indicate that chemically denser material (dregs) at the base of the mantle would have substantial lateral variations in thickness induced by convection of the overlying mantle, and might well form discontinuous aggregations below mantle upwellings. A model with a density contrast of about 2 per cent and an initial uniform thickness of the denser layer of 100 km yields a discontinuous distribution with maximum thickness 230 km and bottom topography of several kilometers amplitude, in reasonable accord with recent seismological observations of vertical and lateral structure. Heat flux out of the core is probably strongly modulated laterally by mantle convection, while mantle dregs will complicate and possibly amplify this effect. Such modulation may be relevant to long-term (10^7 - 10^8 year) variations in the magnetic field

The effect of depth-dependent viscosity on convective mixing in the mantle and the possible survival of primitive mantle

The effect depth-dependent viscosity has on convective mixing and sampling (or degassing) of primitive mantle beneath ridges is explored in two-dimensional models. Higher relative viscosities in the deep mantle decrease convection velocities and strain rates and prolong the residence time of material in the deep mantle. If the average viscosity of the lower mantle is at least 100 times the viscosity of the upper mantle, then some mantle material could have survived from very early in the earth's history. If, in addition, the depth of degassing under ridges has been less than 75 km, on average over earth history, then helium isotopic systematics are qualitatively consistent with whole mantle convection

Revised shock-wave equations of state for high-pressure phases of rocks and minerals

Shock‐wave data for the high‐pressure phases of a number of rocks and minerals have been reanalyzed using a revised seismic equation of state to constrain the zero‐pressure properties of the high‐pressure phases. The anomalously low values of dK/dP resulting from a previous analysis are thereby removed. The inferred zero‐pressure densities of the high‐pressure phases are reduced by an average of 4%, and the values of the zero‐pressure seismic parameter Φ_0 are reduced by up to 30%. bringing them into approximate agreement with the hypothesis of the molar additivity of Φ. For most of the materials considered, the derived pressure trajectories of density versus the seismic parameter Φ are consistent wth shock‐wave data on such materials as MgO, Al_2O_3, and SiO_2 (stishovite) where no zero‐pressure assumptions are required. Iron‐rich compounds may require further revision. It is demonstrated that in poorly constrained cases the Birch‐Murnaghan equation can produce a singularity in dK/dP at high pressure. Possible crystal structures of the high‐pressure phases are considered using the revised zero‐pressure densities. It seems likely that olivines with less than about 10‐mole % FeO content can transform to a phase significantly denser than the isochemical mixture of oxides, in contrast to olivines with higher iron content. The possibility that electron spin transitions occur in iron‐rich compounds is considered, but no strong evidence has been obtained. The derived zero‐pressure densities of the high‐pressure phases are usually not of sufficient accuracy to distinguish between all alternative structures, but in some cases an alternative structure to that previously chosen is preferred here