is described by the mean anomaly l or equivalently the mean longitude ϵ l ϩ . Collectively these variables are called orbital elements. In the Kepler problem, where a single planet orbits a spherical star, all the elements of the planet except the mean longitude are fixed, which is why the elements are useful quantities. We use the masses and a, e, and i from the JPL ephemeris DE200 (Table 2). 6.

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Chemical heterogeneities are introduced into the mantle by subduction of oceanic lithosphere and possibly by subduction of sediments and delamination of the continental lithosphere. These heterogeneities are blended into the mantle by convective mixing. Chemically distinct heterogeneities are stirred into the surrounding mantle matrix by convective shear, which deforms them, increasing the area of contact between an heterogeneity and the matrix. Final homogenization is accomplished by diffusion, which brings centimetersized heterogeneities into chemical equilibrium with the matrix. Numerical models of mantle convection suggest that it is chaotic. To study the effect of a chaotic flow on mixing, we have generated a space‐filling, two‐dimensional flow using the Lorenz equations in the chaotic regime as a driver. We specify a two‐mode expansion of the stream function, resulting in a flow which oscillates smoothly between one and two cells. Particle paths are chaotic in time and space. Mixing is very rapid; the mixing time for heterogeneities with an initial size of 6 km is 240 m.y. for layered mantle convection; the corresponding mixing time for whole mantle convection is 960 m.y. The source region for mid‐ocean ridge basalts has a “marble cake” structure; it is made up of material which has been processed through ridges, subducted, stretched, and thinned and is either partially or wholly homogenized. The incompletely mixed subducted oceanic crust forms bands of eclogite in the peridotite matrix. These bands range in scale from 6 km to a few centimeters.

Abstract.Numerical simulations of the geodynamo have successfully represented many observable characteristics of the geomagnetic field, yielding insight into the fundamental processes that generate magnetic fields in the Earth's core. Because of limited spatial resolution, however, the diffusivities in numerical dynamo models are much larger than those in the Earth's core, and consequently, questions remain about how realistic these models are. The typical strategy used to address this issue has been to continue to increase the resolution of these quasi-laminar models with increasing computational resources, thus pushing them toward more realistic parameter regimes. We assess which methods are most promising for the next generation of supercomputers, which will offer access to O(10 6 ) processor cores for large problems. Here we report performance and accuracy benchmarks from 15 dynamo codes that employ a range of numerical and parallelization methods. Computational performance is assessed on the basis of weak and strong scaling behavior up to 16,384 processor cores. Extrapolations of our weak scaling results indicate that dynamo codes that employ two-or three-dimensional domain decompositions can perform efficiently on up to ∼ 10 6 processor cores, paving the way for more realistic simulations in the next model generation.