Richard J. O'Connell

Elastic moduli of a cracked solid

Abstract Calculations on the basis of the self-consistent method are made for the elastic moduli of bodies containing randomly distributed flat cracks, with or without fluid in their interiors. General concepts are outlined for arbitrary cracks and explicit derivations together with numerical results are given for elliptic cracks. Parameters are identified which adapt the elliptic-crack results to arbitrary convex crack shapes. Finally, some geometrical relations involving randomly distributed cracks and their traces on cross-sections are presented.

Thermal structure, thickness and composition of continental lithosphere

Abstract Global compilations of surface heat flow data from stable, Precambrian terrains show a statistically significant secular change from 41±11 mW/m 2 in Archean to 55±17 mW/m 2 in Proterozoic regions far removed from Archean cratons. Using the tectonothermal age of the continents coupled with average heat flow for different age provinces yields a mean continental surface heat flow between 47 and 49 mW/m 2 (depending on the average, non-orogenic heat flow assumed for Phanerozoic regions). Compositional models for bulk continental crust that produce this much or more heat flow (i.e., K 2 O>2.3–2.4 wt%) are not consistent with these observations. More rigorous constraints on crust composition cannot be had from heat flow data until the relative contributions to surface heat flow from crust and mantle are better determined and the non-orogenic component of heat flow in the areally extensive Phanerozoic regions (35% of the continents) is determined. We calculate conductive geotherms for 41 mW/m 2 surface heat flow to place limits on the heat production of Archean mantle roots and to evaluate the significance of the pressure–temperature ( P – T ) array for cratonic mantle xenoliths. Widely variable geotherms exist for this surface heat flow, depending on the values of crustal and lithospheric mantle heat production that are adopted. Using the average K content of cratonic peridotite xenoliths (0.15 wt% K 2 O, assuming Th/U=3.9 and K/U=10,000 to give a heat production of 0.093 μW/m 3 ) and a range of reasonable crustal heat production values (i.e., ≥0.5 μW/m 3 ), we calculate geotherms that are so strongly curved they never intersect the mantle adiabat. Thus the average cratonic peridotite is not representative of the heat production of Archean mantle roots. Using our preferred estimate of heat production in the cratonic mantle (0.03 wt% K 2 O, or 0.019 μW/m 3 ) we find that the only geotherms that pass through the xenolith P–T data array are those corresponding to crust having very low heat production ( 2 O). If the lithospheric mantle heat production is higher than our preferred values, the continental crust must have correspondingly lower heat production (i.e., bulk crustal K, Th and U contents lower than that of average Archean granulite facies terrains), which we consider unlikely. If the xenolith P–T data reflect equilibration to a conductive geotherm, then Archean lithosphere is relatively thin (150–200 km, based on intersection of the P–T array with the mantle adiabat) and the primary reason for the lower surface heat flow in Archean regions is decreased crustal heat production, rather than the insulating effects of thick lithospheric roots. On the other hand, if the xenolith P–T points result from frozen-in mineral equilibria or reflect perturbed geotherms associated with magmatism, then the Archean crust can have higher heat producing element concentrations, lithospheric thickness can range to greater depths and the low surface heat flow in Archean cratons may be due to the insulating effects of thick lithospheric roots. An uppermost limit for Archean crustal heat production of 0.77 μW/m 3 is determined from the heat flow systematics.

Internal structure of massive terrestrial planets

Abstract Planetary formation models predict the existence of massive terrestrial planets and experiments are now being designed that should succeed in discovering them and measuring their masses and radii. We calculate internal structures of planets with one to ten times the mass of the Earth (Super-Earths) to obtain scaling laws for total radius, mantle thickness, core size and average density as a function of mass. We explore different compositions and obtain a scaling law of R ∝ M 0.267 – 0.272 for Super-Earths. We also study a second family of planets, Super-Mercuries with masses ranging from one mercury-mass to ten mercury-masses with similar composition to the Earths but with a larger core mass fraction. We explore the effect of surface temperature and core mass fraction on the scaling laws for these planets. The scaling law obtained for the Super-Mercuries is R ∝ M ∼ 0.3 .

Subduction zone dip angles and flow driven by plate motion

Abstract Kinematic models of the large scale flow in the mantle accompanying the observed plate motions are calculated by neglecting thermal buoyancy forces. The large scale flow is therefore determined by the mass flux imposed by the moving plates. The energy and momentum equations decouple, and with the assumption of a radially symmetric Newtonian viscosity, the flow accompanying the plate motions can be obtained using harmonic analysis and propagator matrices. The resulting flow models predict remarkably well the observed dips of subducted slabs if the flow extends into the lower mantle. The plates drag along a thick boundary layer which should be included in models of the heating of subducted slabs.

Convection in three dimensions with surface plates: Generation of toroidal flow

This work presents numerical calculations of mantle convection that incorporate some of the basic observational constraints imposed by plate tectonics. The model is three-dimensional and includes surface plates; it allows plate velocity to change dynamically according to the forces which result from convection. We show that plates are an effective means of introducing a toroidal component into the flow field. After initial transients the plate motion is nearly parallel to transform faults and in the direction that tends to minimizes the toroidal flow field. The toroidal field decays with depth from its value at the surface; the poloidal field is relatively constant throughout the layer but falls off slightly at the top and bottom boundaries. Layered viscosity increasing with depth causes the toroidal field to decay more rapidly, effectively confining it to the upper, low-viscosity layer. The effect of viscosity layering on the poloidal field is relatively small, which we attribute to its generation by temperature variations distributed throughout the system. The generation of toroidal flow by surface plates would seem to account for the observed nearly equal energy of toroidal and poloidal fields of plate motions on the Earth. A low-viscosity region in the upper mantle will cause the toroidal flow to decay significantly before reaching the lower mantle. The resulting concentration of toroidal flow in the upper mantle may result in more thorough mixing there and account for some of the geochemical and isotopic differences proposed to exist between the upper and lower mantles.

INEVITABILITY OF PLATE TECTONICS ON SUPER-EARTHS

The recent discovery of super-Earths (masses ≤ ) has initiated a discussion about conditions for habitable 10 M worlds. Among these is the mode of convection, which influences a planet’s thermal evolution and surface conditions. On Earth, plate tectonics has been proposed as a necessary condition for life. Here we show that super-Earths will also have plate tectonics. We demonstrate that as planetary mass increases, the shear stress available to overcome resistance to plate motion increases while the plate thickness decreases, thereby enhancing plate weakness. These effects contribute favorably to the subduction of the lithosphere, an essential component of plate tectonics. Moreover, uncertainties in achieving plate tectonics in the 1 regime disappear as mass M increases: super-Earths, even if dry, will exhibit plate tectonic behavior. Subject headings: Earth — planetary systems — planets and satellites: general

The development of slabs in the upper mantle: Insights from numerical and laboratory experiments

We have performed numerical and laboratory experiments to model subduction of oceanic lithosphere in the upper mantle from its beginnings as a gravitational instability to the fully developed slab. A two-dimensional finite element code is applied to model Newtonian creep in the numerical experiments. Scaled analog media are used in the laboratory, a sand mixture models the brittle crust, silicone putty simulates creep in the lower crust and mantle lithosphere, and glucose syrup is the asthenosphere analog. Both model approaches show similar results and reproduce first-order observations of the subduction process in nature based on density and viscosity heterogeneities in a Stokes flow model. Subduction nucleates slowly and a pronounced slab forms only when the viscosity contrast between oceanic plate and mantle is below a threshold. We find that the subduction velocity and angle are time-dependent and increase roughly exponentially over tens of millions of years before the slab reaches the 670-km discontinuity. The style of subduction is controlled by the prescribed velocity of convergence, the density contrast between the plates, and the viscosity contrast between the oceanic plate and the mantle. These factors can be combined in the buoyancy number F which expresses the ratio between driving slab pull and resisting viscous dissipation in the oceanic plate. Variations in F control the stress in the plates, the speed and the dip of subduction, and the rate of trench retreat, reproducing the contrasting styles of subduction observed in nature. The subduction rate is strongly influenced by the work of bending the lithosphere as it subducts.

On the scale of mantle convection

Abstract Observational evidence from glacial rebound and gravity anomalies shows that the mantle has a relatively uniform viscosity. Such a structure is consistent with the effects of temperature and pressure on microscopic mechanisms for plastic flow of solids. Thus the asthenosphere may extend to the base of the mantle. Mantle phase changes do not inhibit mantle-wide convection, and any compositional differences between the upper and lower mantle are not resolved at present. Thus there is no reason not to expect convection extending into the lower mantle. The stress state of descending slabs is consistent with mantle-wide convection, and although the slabs may encounter an increased resistance to sinking at ∼700 km depth there is no evidence that they are stopped by the increased resistance. Seismic heterogeneity in the lower mantle suggests large-scale convection, which is perhaps driven by heat from the earths core. Plate motions may interact with large-scale flow in the mantle, and are probably not completely decoupled from flow in the lower mantle. Lateral variations in flow properties in the upper mantle may be more important than vertical variations, and may strongly influence plate motions.

THERMAL CONSTRAINTS ON THE SURVIVAL OF PRIMITIVE BLOBS IN THE LOWER MANTLE

Geochemical models have frequently divided the mantle into depleted upper and undepleted lower mantle reservoirs, usually taken as indication for a layered style of convection. This is difficult to reconcile with seismological and geodynamical evidence for substantial mass flux between lower and upper mantle. Various models have been proposed to jointly interpret the evidence, including that of G.F. Davies [J. Geophys. Res. 89 (1984) 6017‐6040] in which the author suggested that lumps of primitive material may exist in the lower mantle, representing reservoirs for undepleted basalts. Mixing calculations have suggested, however, that such blobs could not survive 4 Ga of convection. Calculations by M. Manga [Geophys. Res. Lett. 23 (1996) 403‐406] on the other hand showed that high-viscosity blobs could persist in convective cells for geologically long times without being substantially deformed and mixed with the surrounding flow. We investigate a blob model of convection based on these ideas and consider dynamical, thermal, geochemical and rheological consequences. The radiogenic heat production in the primitive blobs would lead to higher temperatures. However, these would be modest (1T < 300 K) for sufficiently small blobs (radius<800 km). The resulting thermal buoyancy can be offset by a small intrinsic density excess (<1%) so that blob material is hidden from the ridges but sampled by rising plumes. To satisfy geochemical constraints, blobs would have to fill 30% to 65% of the mantle (less if they are taken to be enriched rather than primitive). Thermal considerations require that they be surrounded by depleted material of lower viscosity that would convectively transport heat to the surface. The thermal decrease in blob viscosity would be about one order of magnitude but constrained to the interior; the stiffer ‘shell’ can then be expected to control the dynamical mixing behavior. On average, the viscosity of the lower mantle would be increased by the presence of the blobs; if they were 100 times more viscous than the surrounding mantle the net effect would be to increase the effective viscosity approximately

Ablative subduction - A two-sided alternative to the conventional subduction model

Subduction is conventionally modelled as a convergence and overlapping of two semirigid plates; such a convergence is one-sided, in that one lithospheric plate descends while the other remains on the surface. We suggest an alternative model, one based on the dynamics of fluids. In this model, the viscous lower lithosphere flows downward, and the brittle upper lithosphere deforms in passive response. This process is potentially double-sided, for we find that even a buoyant plate can be dragged downward by a dense, descending neighbor. Thus an apparent overriding plate may be worn away by a process of viscous ablation, with the rate of ablation a function of plate buoyancy. We call this process “ablative subduction”. Ablative subduction allows us to simply interpret observations concerning slab profiles, interplate seismicity, back arc tectonics, and complex processes such as double subduction and subduction polarity reversal. In performing experiments modelling the evolution of simple, fluid “slabs”, we find that slab profile is strongly influenced by ablation in the overriding plate. When ablation is weak, as when a buoyant continent borders the trench, deformable slabs adopt shallow, Andean-style profiles. These profiles develop over time from an initially steep shape. More vigorous ablation rates, as might occur in an ocean-ocean convergence, yield steeper, Marianas-style profiles. Thus differences between Marianas-style and Andean-style slabs may result from differences in ablation rate and subduction duration. The occurrence of ablation might not be easily detected at depth, because material from subducting and ablating plates adhere closely as a single slab. These slabs show downdip deformations that are consistent with seismic focal mechanisms. Plate deformations associated with ablative subduction are also consistent with observed patterns of seismicity. In particular, the relative aseismicity of Marianas-style plate boundaries is consistent with the ablative model; we may thus explain how plates can converge aseismically, without requiring that these plates be decoupled. Ablation should be weakest in Andean-style subduction, although pulses of rapid ablation might lead to episodes of crustal shortening; in oceanic regions, the manifestation of ablation may depend upon tectonic regime. In an extensional area like the Marianas, ablation may be temporally self-limiting, and ablative cycles may explain conflicting observations of tectonic erosion and accretion. In a compressional regime, vigorous ablation might lead to observations of double subduction and subduction reversal. In each case, the behavior of the lower lithosphere is relatively simple; what varies is the rate of ablation and the response of the upper lithosphere.