Engineering topics
Publications and source records attributed to Kaula, W. M..
The Earth is a dynamic system-it has a fluid, mobile atmosphere and oceans, a continually changing distribution of ice, snow, and groundwater, a fluid core undergoing hydromagnetic motion, a mantle undergoing both thermal convection and rebound from glacial loading of the last ice age, and mobile tectonic plates.
We report on numerical simulations designed to understand how the solar system evolved through a winnowing of planetesimals accreeted from the early solar nebula. This sorting process is driven by the energy and angular momentum and continues to the present day. We reconsider the existence and importance of stable niches in the Jupiter/Saturn Zone using greatly improved numerical techniques based on high-order optimized multi-step integration schemes coupled to roundoff error minimizing methods.
It is proposed that subducting tectonic plates can affect the nature of thermal mantle plumes by determining the temperature drop across a plume source layer. The temperature drop affects source layer stability and the morphology of plumes emitted from it. Numerical models are presented to demonstrate how introduction of platelike behavior in a convecting temperature dependent medium, driven by a combination of internal and basal heating, can increase the temperature drop across the lower boundary layer. The temperature drop increases dramatically following introduction of platelike behavior due to formation of a cold temperature inversion above the lower boundary layer. This thermal inversion, induced by deposition of upper boundary layer material to the system base, decays in time, but the temperature drop across the lower boundary layer always remains considerably higher than in models lacking platelike behavior. On the basis of model-inferred boundary layer temperature drops and previous studies of plume dynamics, we argue that generally accepted notions as to the nature of mantle plumes on Earth may hinge on the presence of plates. The implication for Mars and Venus, planets apparently lacking plate tectonics, is that mantle plumes of these planets may differ morphologically from those of Earth. A corollary model-based argument is that as a result of slab-induced thermal inversions above the core mantle boundary the lower most mantle may be subadiabatic, on average (in space and time), if major plate reorganization timescales are less than those acquired to diffuse newly deposited slab material.
We present results from convection models allowing for self-lubrication of downflows. Models impose a line source of chemically light, low viscosity material at the top of a convecting layer of temperature-dependent viscosity material. Low viscosity surface material serves as an analog to hydrated sediment/crust and the high viscosity upper portion of the convecting layer as an analog to mantle lithosphere. Slow near surface motion in the convecting layer entrains low viscosity material into zones of downflow, which has a lubricating effect. Once entrained lubricant is deeper than the cold high viscosity portion of the convecting layer, rapid upper boundary layer overturn occurs and system properties change (e.g., heat flux doubles). This marks transition to a lubricated state. Before and after transition, transport properties are dominantly determined by, respectively, the viscosity of mantle lithosphere and that of interior mantle. Lubricated and nonlubricated states appear as distinct regions in system output space suggesting that exchange between them is akin to a phase transition. That such exchange depends on a near surface lubricant implies that the geodynamics of planets lacking such lubricants may fundamentally differ from that of planets possessing them.
Owen et al. propose that the Ar excess of Venus compared to Earth was caused by impact of a large icy planetesimal from the outer solar system, where temperatures were low enough for Ar to adhere to ice, about 30 K. A body of solar Ar/Si and C/H similar to Pluto and Triton less than 100 km diameter would suffice. However, direct delivery from the Uranus-Neptune zone to Venus would result in a very high approach velocity, causing erosion rather than accretion of volatiles. It would also be an extremely improbable event. Virtually all icy bodies scattered from the Uranus-Neptune zone to the terrestrial zone were strongly perturbed by Jupiter, but even then arrived at Venus (or Earth) at too-high velocities. What is needed is a 'deboost' of bodies scattered inward from Jupiter by Earth and Venus perturbations, prior to impact into Venus. This has low probability, since the influence radii of Venus and Earth for the high velocities are small compared to that of Jupiter for the lower velocities further out. It is desirable that this problem be examined by an integerator that allows for close approaches, such as developed by Levison and Duncan. However, it would be a long and expensive computation compared to explaining short-period comets to accomplish the third 'deboost' phase. But since close encounters dominate in such distant scattering, an Opik algorithm seems the appropriate reconnaissance tool.
Entrainment of lower crust by convective mantle downflows is proposed as a crustal recycling mechanism on Venus. The mechanism is characterized by thin sheets of crust being pulled into the mantle by viscous flow stresses. Finite element models of crust/mantle interaction are used to explore tectonic conditions under which crustal entrainment may occur. The recycling scenarios suggested by the numerical models are analogous to previously studied problems for which analytic and experimental relationships assessing entrainment rates have been derived. We use these relationships to estimate crustal recycling rates on Venus. Estimated rates are largely determined by (1) strain rate at the crust/mantle interface (higher strain rate leads to greater entrainment); and (2) effective viscosity of the lower crust (viscosity closer to that of mantle lithosphere leads to greater entrainment). Reasonable geologic strain rates and available crustal flow laws suggest entrainment can recycle approximately equal 1 cu km of crust per year under favorable conditions.
Effective numerical treatment of multicomponent viscous flow problems involving the advection of sharp interfaces between materials of differing physical properties requires correction techniques to prevent spurious diffusion and dispersion. We develop a particular algorithm, based on modern shock-capture techniques, employing a two-step nonlinear method. The first step involves the global application of a high-order upwind scheme to a hyperbolic advection equation used to model the distribution of distinct material components in a flow field. The second step is corrective and involves the application of a global filter designed to remove dispersion errors that result from the advection of discontinuities (e.g., material interfaces) by high-order, minimally dissipative schemes. The filter introduces no additional diffusion error. Nonuniform viscosity across a material interface is allowed for by the implementation of a compositionally weighted-inverse interface viscosity scheme. The combined method approaches the optimal accuracy of modern shock-capture techniques with a minimal increase in computational time and memory. A key advantage of this method is its simplicity to incorporate into preexisting codes be they finite difference, element, or volume of two or three dimensions.
The horizontal locations of craters on Venus are consistent with randomness. However, (1) randomness does not make crater counts useless for age indications; (2) consistency does not imply necessity or optimality; and (3) horizontal location is not the only reference frame against which to test models. Re (1), the apparent smallness of resurfacing areas means that a region on the order of one percent of the planet with a typical number of craters, 5-15, will have a range of feature ages of several 100 My. Re (2), models of resurfacing somewhat similar to Earth's can be found that are also consistent and more optimal than random: i.e., resurfacing occurring in clusters, that arise and die away in lime intervals on the order of 50 My. These agree with the observation that there are more areas of high crater density, and fewer of moderate density, than optimal for random. Re (3), 799 crater elevations were tested; there are more at low elevations and fewer at high elevations than optimal for random: i.e., 54.6 percent below the median. Only one of 40 random sets of 799 was as extreme.
The geoid and topography heights of Atla Regio and Beta Regio, both peaks and slopes, appear explicable as steady-state plumes, if non-linear viscosity eta(Tau, epsilon) is taken into account. Strongly constrained by the data are an effective plume depth of about 700 km, with a temperature anomaly thereat of about 30 degrees, leading to more than 400 degrees at the plume head. Also well constrained is the combination Q(eta)/s(sup 4)(sub 0) = (volume flow rate)(viscosity)/(plume radius): about 11 Pa/m/sec. The topographic slopes dh/ds constrain the combination Q/A, where A is the thickness of the spreading layer, since the slope varies inversely with velocity. The geoid slopes dN/ds require enhancement of the deeper flow, as expected from non-linear viscosity. The Beta data are best fit by Q = 500 m(sup 3)/sec and A equals 140 km; the Atla, by Q equals 440 m(exp 3)/sec and A equals 260 km. The dynamic contribution to the topographic slope is minor.
The upper boundary layer of Venus is comprised of at least two distinct chemical components, mantle and crust. Fluid dynamical models of convection within Venus' mantle were primarily of the thermal boundary layer type. Models assessing the ability of convective mantle flows to deform the crust were undertaken, but models exploring the effects of a variable thickness crust on mantle convection were largely lacking. A Venusian crust of variable thickness could couple back into, and alter, the mantle flow patterns that helped create it, leading to deformation mechanisms not predicted by purely thermal boundary layer convection models. This possibility is explored through a finite element model of thermal/chemical boundary layer convection. Model results suggest that a crust of variable thickness can serve as a mantle flow driver by perturbing lateral temperature gradients in the upper mantle. Resulting mantle flow is driven by the combination of free convective and nonuniform crustal distribution. This combination can lead to a flow instability manifest in the occurrence of episodic mantle lithosphere subduction initiated at the periphery of a crustal plateau. The ability of a light, near surface, chemical layer to potentially alter mantle flow patterns suggest that mantle convection and the creation and/or deformation of such a chemical layer may be highly nonseparable problems on time scales of 10(exp 8) years.
Models incorporating plate-like behavior, i.e., near uniform surface velocity and deformation concentrated at plate boundaries, into a convective system, heated by a mix of internal and basal heating and allowing for temperature dependent viscosity, were constructed and compared to similar models not possessing plate-like behavior. The simplified numerical models are used to explore how plate-like behavior in a convective system can effect the lower boundary layer from which thermal plumes form. A principal conclusion is that plate-like behavior can significantly increase the temperature drop across the lower thermal boundary layer. This temperature drop affects the morphology of plumes by determining the viscosity drop across the boundary layer. Model results suggest that plumes on planets possessing plate-like behavior, e.g., the Earth, may differ in morphologic type from plumes on planets not possessing plate-like behavior, e.g., Venus and Mars.
The thermal state of the Earth at the time relevant to formation of a magma ocean was dominated by the great impact that created the Moon. As shown in computer experiments, the iron in the impacting bodies quickly sank to the core of the proto-Earth, while a significant fraction of silicates was pushed far enough out beyond the geosynchronous limit to constitute the main material of the Moon. Most of any atmosphere would have been pushed aside, rather than being expelled in the impact. However, the energy remaining in the material not going to the core or expelled was still sufficient to raise its temperature some 1000's of degrees, enough to vaporize silicates and to generate a strong 'planetary wind': a hydrodynamic expansion carrying with it virtually all volatiles plus appreciable silicates. This expansion was violent and uneven in its most energetic stage, but probably the resulting magma ocean was global. The duration, until cooling, was sufficient for silicates to condense to melt and the duration was probably short. Comparison of the Earth and Venus indicates that the great impact was extraordinarily effective in removing volatiles from the proto-Earth; in particular, the enormous differences in primordial inert gases between the planets demand a catastrophic difference in origin circumstances. On the other hand, the comparison limits the amount of silicates lost by the Earth to a rather minor fraction; most of that expelled in the wind must have condensed soon enough for the silicate to fall back to Earth or be swept up by the proto-Moon. So the Earth was left with a magma ocean. The question is whether sufficient water was retained to constitute a steam atmosphere. Probably not, but unknowns affecting this question are the efficiencies of outgassing in great impacts and in subsequent convective churnings deep in the mantle. During the stage when mantle convection is turbulent, an appreciable fraction of volatiles were also retained at depth, perhaps in some mineral phases not yet well-defined. We still have primordial helium being outgassed.
Two of the most important constraints are known from Pioneer Venus data: the lack of a system of spreading rises, indicating distributed deformation rather than plate tectonics; and the high gravity/topography ratio, indicating the absence of an asthenosphere. In addition, the high depth/diameter ratios of craters on Venus indicate that Venus probably has no more crust than Earth. The problems of the character of tectonics and crustal formation and recycling are closely coupled. Venus appears to lack a recycling mechanism as effective as subduction, but may also have a low rate of crustal differentiation because of a mantle convection pattern that is more distributed, less concentrated, than Earth's. Distributed convection, coupled with the nonlinear dependence of volcanism on heat flow, would lead to much less magmatism, despite only moderately less heat flow, compared to Earth. The plausible reason for this difference in convective style is the absence of water in the upper mantle of Venus. We have applied finite element modeling to problems of the interaction of mantle convection and crust on Venus. The main emphasis has been on the tectonic evolution of Ishtar Terra, as the consequence of convergent mantle flow. The early stage evolution is primarily mechanical, with crust being piled up on the down-stream side. Then the downflow migrates away from the center. In the later stages, after more than 100 m.y., thermal effects develop due to the insulating influence of the thickened crust. An important feature of this modeling is the entrainment of some crustal material in downflows. An important general theme in both convergent and divergent flows is that of mixing vs. stratification. Models of multicomponent solid-state flow obtain that lower-density crustal material can be entrained and recycled, provided that the ration of low-density to high-density material is small enough (as in subducted slabs on Earth). The same considerations should apply in upflows; a small percent of partial melt may be carried along with its matrix and never escape to the surface. Models that assume melt automatically rising to the crust and no entrainment or other mechanism of recycling lower-density material obtain oscillatory behavior, because it takes a long time for heat to build up enough to overcome a Mg-rich low-density residuum. However, these models develop much thicker crust than consistent with estimates from crater depth/diameter ratios.
A three dimensional Cartesian code was developed to model molecular cloud collapse and protostellar formation. The basic Eulerian scheme employs spatially second-order accurate, finite difference methods to advance the fluid variables in time. The hydrodynamic equations governing the collapse include the effects of self-gravity, rotation, and radiative transfer. Supplementary equations include Poisson's equation which relates the gravitational potential to the density distribution and a mean intensity equation derived with the Eddington approximation for radiative transfer. The gas pressures and internal energies are determined from equations of state derived for a molecular cloud composed of hydrogen, helium, and heavier elements.
It is proposed that western Ishtar Terra formed due to compression and crustal thickening above a cylindrical mantle downwelling. A model for crustal deformation due to downwelling successfully reproduces many observed characteristics of western Ishtar. Although axisymmetric downwelling occur in numerical models of constant-viscosity mantle convection, there is no evidence for their existence in earth's mantle, where downwellings are sheet-like. Either modes of downflow in Venus and earth are fundamentally different, or differences in near-surface conditions and material behavior selectively emphasize surface expressions of the different downwelling modes.
The present experiment with test particles placed in orbits at uniform intervals between 5.7 and 8.8 AU, with randomly distributed eccentricities between 0.0 and 0.02 and inclinations between 0.0 and 0.063 rad, leads to the perception that no stable niches exist in the Jupiter-Saturn zone other than the Lagrangian points of Jupiter. These points are populated by at least 26 Trojan asteroids. The primary implication of these results for future investigations of the Saturn-Uranus-Neptune zone is that experimental design should be as selective as the dynamics of the planetary system have been shown to be.
Aspects of star formation are considered, taking into account those observations and models which have a relation to the formation of the solar system. Attention is given to the galactic structure, the stellar properties, planetary indications, interstellar clouds, observations of forming stars, the conditions for cloud collapse, models of star formation, the modeling of secondary features, and aspects of dynamical evolution. Aspects of cosmochemistry are also explored. Questions of chronology are discussed along with nuclide variations, chemical variations among chondritic meteorites, and chemical variations among differentiated objects. A description of planet formation is provided, taking into consideration, gas-dust interaction, sun and accretion disk dynamics, planetesimal swarms, planet-disk interactions, planetary systems, terrestrial planet formation, and gaseous protoplanet contraction.
The dynamics of iron, its thermal state and its phase in the accreting Earth probably played a major role in the Earth's early thermal evolution. Plausible impact thermal histories make it possible that pure iron was molten in the accreting Earth after it was about 10% grown. Hence, iron eutectic alloys (FeS, FeO) certainly were. Additionally, the initial temperature of the core is an important constraint on the secular cooling of the early Earth and on the strength of the early geodynamo. Whether iron is solid or molten would influence geochemical equilibria in the upper and lower mantle; the mode of core formation, by spherical or near-spherical blobs, stalk-like instabilities, or something more catastrophic would influence the partitioning of siderophiles between silicate and iron phases. Early descent of iron (during accretion) favors partitioning according to low-pressure phase equilibria, whereas late descent favors higher pressure. The later core formation occurs, the greater the heat pulse, due to the strong dependence of gravitational potential energy on planetary radius. The heat may homogenize the mantle if core formation is global; otherwise, heterogeneity of iron differentiation may leave some of the pre-archean mantle unaffected. The larger the chunks of proto-core (and hence smaller surface/volume ratios) the greater the heterogeneity.