chaotic axisymmetrical spherical convection and large scale mantle circulation abstract this paper presents a study of high rayleigh number up to 200 times supercritical axisymmetrical convection in a spherical shell with an aspect ratio relevant for the earth s lower mantle both bottom heated and internal heated cases have been considered computations have been carried out for an infinite prandtl number isoviscous fluid with free slip isothermal boundary conditions the first part of the paper is devoted to the influence of the resolution on the accuracy of the numerical results it is shown that the resolution strongly influences the onset of time dependence recent methods of non linear physics have been used to prove that the time dependence and the chaotic behaviors of the solutions are real ones from these results we can confirm that convection is chaotic in this particular geometry even for rayleigh numbers 200 times critical aperiodic boundary layer instabilities are found to be incapable of breaking up the large scale flow owing to the shear of the global circulation spectral analysis of the power associated with the thermal anomalies shows that there is an upward cascade of energy due to small scale chaotic instabilities from l 2 to l 4 6 at the bottom boundary in agreement with new seismic observations at the core mantle boundary 1 3

mantle convection models featuring plate tectonic behavior an overview of methods and progress abstract arguably the presence of plate tectonic type surface motion for periods that endure over hundreds of millions of years is the primary feature a mantle convection model must possess in order to be considered earth like from the early days of mantle dynamics modeling research has been dedicated to understanding how mantle convection produces the first order observations of plate tectonics as well as how the plates and deep mantle interact fledgling studies of the effect of plates on the mantle recognized the ability of imposed plate scale surface motion to influence global temperatures and heat flux and organize convective planform later studies featuring model plates with dynamically determined velocities discovered that the interaction between convection and the plates could result in cyclic plate motion patterns and other time dependent behavior that was not manifested in systems in which dynamic plates were absent focussing on different aspects of system realism with respect to terrestrial mantle convection has spawned multiple approaches for modeling convection with dynamic integrated plates in broadest terms the two main approaches can be categorized as rheological modeling methods and methods utilizing evolving surface boundary conditions over the past dozen years studies focussing on the former approach have steadily made progress in modeling the self generation of plate tectonics from convection dynamics continual advances have been encouraging and a consensus is beginning to form regarding the necessary requirements for obtaining the primary elements of plate like surface motion however despite significant progress the generation of plates over long periods has not yet been modeled with earth like convective vigor in contrast models utilizing dynamically determined boundary conditions to achieve plate like surface motion have relatively little difficulty with emulating terrestrial convective vigor or simulations of billions of years instead their weakness is more fundamental they can only provide insight into the reciprocating dynamics of the mantle and plates once the existence of the plates is assumed and they cannot model any aspects of the dynamics responsible for the origin of the plates this paper briefly reviews the evolution of mantle convection models featuring plates and examines the progress that has been made in our understanding of the feedback between the mantle and plate tectonics through the use of both rheological and boundary condition modeling methods common findings recent advances and unbridged problems are identified and discussed

steady plumes in viscously stratified vigorously convecting three dimensional numerical mantle convection models with mobile plates 1 we present results from three dimensional 3 d cartesian geometry computations featuring vigorous high rayleigh number mantle convection in a system incorporating stiff tectonic plates with dynamically determined time dependent velocities we track plume related hot spot motion while calculating associated synthetic hot spot tracks allowing us to estimate plume migration rates relative to plate motion rates plate like surface motion is achieved explicitly by specifying both the plate geometry and rigidity however plate velocities evolve dynamically in response to the buoyancy distribution within the convecting system including buoyancy within the plate itself we find that convection is characterized by downwelling sheets and varying numbers of 3 d upwelling structures when the lower mantle to upper mantle viscosity ratio is varied from 9 to 90 as the lower mantle viscosity is increased relative to the upper mantle viscosity the upwellings in the calculations evolve into vigorous active plumes characterized by long narrow thermal conduits with broad disk like heads the total number shape and persistence of the plumes are affected by the specified viscosity stratification in the calculations our findings indicate that hot spots associated with mantle plumes drift by smaller distances in comparable amounts of time when the lower mantle viscosity is increased relative to the upper mantle viscosity in cases with a high lower mantle to upper mantle viscosity ratio the depth of the flow aligned with plate motion is diminished and the plate scale return flow is integrated into a deep unorganized sluggish layer as a result plume conduit morphology in the lower mantle of such calculations is less influenced by plate motion than in calculations with a small difference between upper mantle and lower mantle viscosity for lower mantle viscosities 30 times and 90 times greater than the upper mantle viscosity the lower mantle velocity field is dominated locally by active plumes anchored in the lower thermal boundary layer of the convecting system and plume migration rates are typically only about 10 of the overriding plate velocity in such cases we also find that plume related hot spots persist for periods that we estimate to be between 0 55 and 1 1 plume transit times

time dependent convection in the earth s mantle abstract the range of rayleigh numbers accessible to 2 dimensional numerical models of mantle convection has been increased by almost two orders of magnitude through the use of high resolution numerical grids with up to 200 intervals in each of the horizontal and vertical directions the range of rayleigh numbers studied is believed to encompass conditions which would have existed in a hot archaean mantle although the models are highly idealized in that they neglect the compressibility and variations of viscosity in the earth s mantle they suggest that during the archaean and earlier boundary layer instabilities developed more frequently than at present and did so with a scale and separation comparable to the thickness of the thin thermal boundary layers rather than to the depth of the convecting layer itself the rising and sinking plumes which develop from these instabilities drive transient flows which nevertheless have a horizontal scale comparable to the depth of the convecting layer thus although surface motions during the archaean may have been more variable than at present in terms of speed and direction they might have occurred on a horizontal scale which is similar to that of present day plate motion the numerical models indicate that motions at the upper surface of time dependent convection cells are 1 predominantly controlled by transient sinking plumes and 2 converge directly towards the source region of the sinking plumes on the other hand rising plumes have a more diffuse influence on the upper surface and surface divergence does not always occur immediately above a rising plume if oceanic plates on earth constitute the upper surface of mantle wide convection cells the more pronounced geophysical signature of subduction zones compared to oceanic ridges may be simply due to the fact that buoyancy is most concentrated at the earth s surface in the form of sinking plumes the total surface heat flow is found to fluctuate significantly in the time dependent convection models since estimates of the present day heat loss from the earth s interior are at best instantaneous measurements it is possible that these estimates are not representative in a time averaged sense of the present thermal state of the planetary interior

some geodynamical effects of anisotropic viscosity summary rheological anisotropy in the mantle may arise from various causes the most important are the preferred orientation of single crystals and the orientation of streaks of eclogitic material in a marble cake mantle in both cases the orientation can be achieved by the flow kinematics and in steady state convection the planes of easy slip will eventually be orientated parallel to the stream lines the dynamic equations for a 2 d incompressible anisotropic viscous fluid are derived and some geodynamic consequences are studied a difference of the order 10 100 between shear and normal viscosity is considered post glacial rebound would be affected by the mechanical properties in deeper parts of the mantle the observed uplift history near the ice margin has been taken as argument against strong viscosity stratification in the mantle it is shown here that with strong viscous anisotropy at least in the upper mantle the uplift history is compatible with a significant increase in viscosity between upper and lower mantle the topographic and geoid signal can show a spatial offset from a generating mass anomaly in the mantle this may provide the opportunity for in situ observation of anisotropic flow properties in the mantle in numerical case studies of steady state convection streamline orientated anisotropy is considered at high rayleigh number for bottom heated convection the formation of velocity boundary layers is observed with no flow in the stagnant core of the convection cell with internal heating or strong pressure and temperature dependence of viscosity this is no longer found but in the latter case anisotropy enhances the channelling of flow into lowviscosity regions e g hot rising plumes a few experiments have been performed for time dependent convection the results suggest that the onset of time dependent boundary layer instabilities may be somewhat retarded but would not be strongly suppressed by anisotropy strong time dependence may destroy any large scale coherence in the orientation of anisotropy and thus render the mantle to appear viscously isotropic

geoid and topography of earth like planets a comparison between compressible and incompressible models for different rheologies abstract a systematic study of 2d axisymmetric spherical shell models of compressible and incompressible mantle convection with constant and variable viscosity and constant and depth dependent thermodynamic properties is presented to account for compressibility effects we employ the anelastic liquid approximation in the case of variable viscosity an arrhenius law with strongly temperature and pressure dependent viscosity is considered we show that assuming compressible convection with depth dependent thermodynamic properties strongly influence the geoid undulations using compressible convection with constant thermodynamic properties is physically inconsistent and may lead to spurious results for the geoid and convection pattern in addition we examine the impact of compressibility as well as different rheologies on the power law relation that connects the nusselt number to the rayleigh number we discover that the power law index of the nu ra relationship is controlled by the rheology independent of which approximation is used instead the bound of this relation is controlled by a combination of different approximation and rheology

the evolution of hadean eoarchaean geodynamics abstract geodynamic modelling of hadean eoarchaean tectonics typically requires higher rates of internal heat production and higher mantle temperatures in models that possess temperature dependent viscosity and a yield criterion in such models under hadean conditions for a wide range of geodynamic configurations and modelling approaches subduction has a propensity to fail this has led to the suggestion that the predominant tectonic regime in the hadean was stagnant lid convection with intermittent recycling events various lines of evidence support this suggestion from i the long mixing time of mantle isotopic anomalies or compositional heterogeneities such as 142nd 182w and platinum group elements to ii the long residence time of the hadean protolith to the jack hills zircons and iii thermal evolution models which typically require lower heat flux in the past to avoid the archaean thermal catastrophe the framework provided by stagnant lid or episodic overturn convection iv provides an explanation for the formation of early archaean ttgs and greenstones and v explains the interleaving arc plume sequence observed in many archaean terranes suggesting subduction initiation events may have been common increasing their preservation potential implications include a low magnetic field strength in the hadean which is consistent with emerging paleointensity data from these times

whole mantle convection and the thermal evolution of the earth abstract three recent results in the fields of geophysics and geochemistry have combined to rejuvenate the study of the earth s thermal history the first and perhaps most important of these is the new understanding of the planetary accretion process which has been realized as a consequence of work by v s safronov and others this work shows that the terrestrial planets accrete on a time scale of 10 8 y by the accumulation of planetesimal sized objects and that they probably form sufficiently hot to induce immediate differentiation of iron from silicates the second important recent contribution has been from geochemistry and concerns the net complement of radioactivity of the earth as a whole although the net complement of radioactive elements has usually been inferred on the basis of the assumption of equilibrium with the surface heat flow a priori analysis based upon elemental condensation temperatures in the primitive solar nebula yields a model in which the net radioactivity is at least a factor of two less than would be required to explain the mean surface heat flow there are good reasons to believe that this model is valid for the earth the third result has been the successful development of a technique for incorporating the influence of convective heat transfer in a thermal evolution model which does not require explicit solution of the dynamical equations this paper begins with an analysis of the mantle convective circulation which is required to understand plate tectonics and continental drift an analysis which presents the arguments for and against the crucial notion that the main circulation has a depth scale equal to the mantle thickness using the idea of parameterized convection the radial heat transfer by the mantle convective circulation is included in a thermal history model of the earth this model turns out to contain within it an interesting explanation of the fact that no crustal rocks with ages in excess of 3 8 ga have ever been found an observation that has heretofore proven enigmatic

crystal settling in a vigorously converting magma chamber there has been much debate concerning the mechanism of fractional crystallization in magma chambers the traditional hypothesis of crystal settling has been widely replaced by the concept of in situ crystallization coupled with compositional convection observations from layered intrusions however are equivocal1 4 doubts have been raised about crystal settling on theoretical grounds because convective velocities in magma chambers are often much greater than the crystal settling velocities predicted by stokes law5 but there has been no experimental study of crystal settling in such vigorous convection here we present physical considerations and laboratory experiments which show that the phenomenon of particle settling in these conditions can be accounted for by a simple theory application of this theory to crystal settling in magma chambers suggests that crystal settling may be an efficient differentiation mechanism at least in basaltic magma chambers despite large convective velocities

the role of the heating mode of the mantle in intermittent reorganization of the plate velocity field summary the geological record indicates that stages of relatively steady plate motion have been punctuated by comparatively brief periods in which plate velocities have reorganized the distribution of buoyancy sources in the mantle has generally been regarded as evolving too slowly to explain these rapid transitions in plate velocity we investigate the feedback between mantle convection and plate velocity using 2 d and 3 d mantle convection models that incorporate mobile dynamic plates we focus on the influence of internal heating in the mantle and consider the effect of mantle viscosity stratification and different plate geometries on the plate velocity time dependence as either the rayleigh number or the internal heating rate is increased to magnitudes approaching mantle values the record of the plate motion from our calculations becomes characterized by intermittent changes in direction this behaviour is a result of the influence of plates on heat loss from the inherently unsteady internally heated convecting system plate motion instills a pattern of organization on the underlying convection that reflects the plate geometry and results in the formation of sheet like downwelling structures at convergent plate boundaries in both 2 d and 3 d calculations in contrast upwellings in 3 d models are not sheet like the role of the sheet like downwellings is critical in the observed episodic reorganization of the plate velocities warm material below the plates is entrained by plate motion into regions enveloping the downwelling sheets during periods of fairly steady plate motion buoyancy associated with the build up of heat around the downwelling sheets leads to the creation of an unstable convection pattern this build up of heat is dramatic in calculations with mantle like internal heating rates and resists continued long term plate motion towards mature downwellings when there are limitations on the degree of freedom of the direction of plate movement such as in 2 d models these effects become even more pronounced accordingly the effect of plates on mean global thermal quantities is more dramatic in 2 d calculations than it is in 3 d calculations nevertheless 3 d calculations incorporating plates of different sizes do exhibit rapid reorganizations in their convection patterns as the pull of young slablike features supersedes the pull of mature downwelling sheets we compare the timing and frequency of the reorganization events in our calculations with the general characteristics of plate motions determined from plate reconstruction studies

time dependent convection with non newtonian viscosity we have used a numerical model of bottom heated two dimensional convection in boxes of aspect ratios 2 5 and 4 0 to study the differences in time dependent convection between newtonian and stress dependent viscosity the onset of time dependence due to boundary layer instability is found at approximately the same effective rayleigh number for both rheologies however with increasing rayleigh number the temporal and spatial fluctuations in the flow field become much more pronounced with non newtonian rheology also the tendency for breakup of long cells into smaller ones is stronger the presence of stress dependent rheology can cause long wavelength lateral viscosity variations up to an order of magnitude this result could have strong implications for the interpretation of mantle viscosity from postglacial rebound

two dimensional mantle convection simulations using an internal state variable model the role of a history dependent rheology on mantle convection summary we apply the bammann inelastic internal state variable model biisv to a mantle convection code terra2d to investigate the influence of a history dependent solid mechanics model on mantle convection we compare and contrast the general purpose biisv model to the commonly used power law model we implemented the biisv model using a radial return algorithm and tested it against previously published mantle convection simulation results for verification model constants for the biisv are used based on experimental stress strain behaviour found in the literature after implementation we give illustrative simulation examples were the biisv produces hardened areas on the cold thermal boundary layer that the power law model cannot produce the hardened boundary layers divert material downward giving a plausible reason for the current subduction zones that are present on the earth

the deep structure of ocean ridges in a convecting mantle abstract plate tectonic models have frequently assumed that mid ocean ridges are the surface expression of the hot rising limb of a convection cell measured half spreading rates show that in any reference frame the horizontal transverse velocity of some ridges is comparable to the individual plate velocities if the above assumption is correct this observation implies that the thermal structure beneath the ridge can adjust rapidly to the changing surface configuration of the plates numerical experiments in a two dimensional convecting layer of constant viscosity fluid show that such a rapid adjustment is probably unrealistic a symmetrically spreading ridge moving horizontally is simulated by specifying a velocity boundary condition on the upper surface of the layer the other boundary conditions are uncertain so several different combinations are used in the experiments including periodic or reflecting lateral boundaries and stress free or no slip lower boundary in all the experiments the rising sheets of the convection cells are anchored firmly in the mean horizontal flow of the layer unless the mean flow coincides approximately with the ridge velocity the ridge quickly moves away from a rising sheet which is initially beneath it if the aspect ratio width depth of the convection cells is significantly greater than one this conclusion is even more evident in the earth it seems unlikely that there is a systematic deep of the order of the convecting layer thickness structure beneath the worldwide network of ocean ridges

mantle convection with spherical effects results of a similarity theory for spherical mantle convection are presented the single mode mean field equations are analyzed for convection which is so vigorous that temperature disturbances become localized in thin thermal boundary layers our purpose is to study effects of spherical geometry density interfaces heat source distribution and cell size steady state solutions are found for isoviscous spherical shells in which the field of motion is spatially periodic in a single spherical harmonic degree calculations are carried out over the range 2 l 40 and for various fractions of internal versus base heating three configurations are examined 1 convection in a single layer of cells extending through the whole mantle 2 convection in two layers separated by a density interface at 670 km depth and 3 convection in a single layer terminating at 670 km results of these calculations are used to give estimates of surface horizontal velocities in terms of the heat loss viscosity stratification amount of internal heating and depth of circulation the surface velocity is most strongly affected by the thickness of the convecting shell deep mantle convection can achieve surface velocities which agree with observed plate speeds while convection restricted to the upper mantle does not at least on the scale of the major plates the temperature distribution is strongly affected by the spherical geometry and by the presence of density interfaces the principal difference between convection in one and two layers is that the latter produces a hot lower mantle while the former produces a warm one

surface deflections due to transient subduction in a convecting mantle abstract transient surface deflections associated with non continuous subduction are studied through the use of a sequence of idealized numerical models of time dependent mantle convection the major concerns of this study are the magnitude and duration of dynamically supported topographic fluctuations resulting from either the initiation or termination of a subduction zone at the earth s surface the former is modelled by prescribing an initial temperature structure which mimics the presence of a cold slab of lithosphere immersed in a warm uniform temperature mantle while the latter is induced by thermally detaching this slab from the upper surface of the convection cell the model slabs are thus defined thermally rather than mechanically and their negative buoyancy induces convective flow in the neighbouring mantle the full hydrodynamic equations governing natural convection are solved numerically in order to follow the evolution of the temperature and velocity fields with time the resulting model flows are therefore dynamically self consistent and differ from previous kinematic models in that the flow velocities are determined at each instant from the evolving thermal field rather than prescribed as boundary conditions as the lithospheric slab sinks into the mantle the induced flow produces normal stresses which in turn result in a broad topographic depression subsequently as the slab surface distance increases the topography rebounds at a rate which is strongly dependent upon the imposed temperature contrast between the slab and the surrounding mantle material assuming that these depressions are filled with sediments if a subsequent episode of subduction were to begin before the initial depression was eliminated new sediments would be superimposed upon the old repeated episodes of subduction at a continental margin may therefore be an important factor contributing to repeated cycles of platform sedimentation

minimum heat flow from the core and thermal evolution of the earth abstract the role of heat flow coming from the core is often overlooked or underestimated in simple models of earth s thermal evolution throughout most of earth s history the mantle must have been extracting from the core at least the amount of heat that is required to operate the geodynamo in view of recent laboratory measurements and theoretical calculations indicating a higher thermal conductivity of iron than previously thought the above constraint has important implications for the thermal history of the earth s mantle in this work we construct a paramaterized mantle convection model that treats both the top and the core mantle boundary heat fluxes according to the boundary layer theory or alternatively employs the model of labrosse 2015 to compute the thermal evolution of the earth s core we show that the core is likely to provide all the missing heat that is necessary in order to avoid the so called thermal catastrophe of the mantle moreover by analyzing the mutual feedback between the core and the mantle we provide the necessary ingredients for obtaining thermal histories that are consistent with the petrological record and have reasonable initial conditions these include a sufficiently high viscosity contrast between the lower and upper mantle whose exact value is sensitive to the activation energy that governs the temperature dependence of the viscosity

formation of seafloor swells by mantle plumes the prominent oceanic hot spots are located at the crest of broad topographic swells on the seafloor available evidence indicates that swells are supported by anomalously thin or altered lithosphere as might result from the intrusion of a mantle plume into the lithosphere a set of experiments have been made to measure the free surface uplift and subsidence produced by intrusion of a low viscosity buoyant plume into a rheological boundary layer a deep layer of strongly thermoviscous sugar solution with a free surface was cooled from above to simulate the lithosphere asthenosphere system low viscosity low density plumes were released at depth the free surface displacement generated by the ascending plume was recorded as a function of time using a proximity measurement system the plumes ascend through the asthenosphere as nearly spherical diapirs and collapse flatten upon intrusion into the base of the lithosphere evolution of the surface topography can be divided into three phases 1 a low axisymmetric rise supported by the plume deep beneath the lithosphere 2 rapid uplift and steepening as the plume reaches the lithosphere base and 3 slow axial subsidence as the plume stagnates within the lithosphere the measured uplifts are compared with the uplift calculated for buoyant plumes in a uniform viscous half space using morgan s image method the agreement is good for thin lithosphere for thick lithosphere the observed is less than the calculated uplift scaling laws have been derived which relate the measured maximum uplift and maximum uplift rate to diapir parameters extrapolation of these to the scale of ocean floor swells gives uplift rates of approximately 1 km in 5 m y by this mechanism the most significant qualitative results of the experiments are that swells formed by interaction of low viscosity mantle plumes and oceanic lithosphere can be characterized by axial symmetry rapid uplift and shallow compensation

hard turbulent thermal convection and thermal evolution of the mantle this article summarizes the results of hard turbulent convection obtained in laboratory experiments and numerical simulations its applications to mantle convection are illustrated by two dimensional numerical solutions to 1 newtonian 2 non newtonian convection and 3 newtonian convection with multiple phase transitions in newtonian mantle convection the transition from soft to hard turbulence is marked by the appearance of disconnected plumes spectral analysis of the time series of the nusselt number reveals the presence of a spectral scaling subrange for hard turbulence but not for soft turbulence in hard turbulence there is correspondence between the spectra in frequency and wavenumber domains the slope of the seismic wave spectra measured from seismology suggests that the mantle convection today is strongly time dependent the transition to hard turbulence takes place at much lower nusselt numbers for non newtonian than for newtonian rheology for the mantle this would have important ramifications non newtonian plumes behave quite differently from newtonian ones in that large curvatures are developed in their trajectories in the hard turbulent regime mantle convection with phase transitions tends to become more layered with increasing rayleigh numbers the style of mantle convection might have changed from a layered to a more whole mantle type of flow with time catastrophic overturns associated with strong gravitational instabilities in the transition zone could be responsible for superplume events

effects of lithospheric rigidity on ocean floor bathymetry and heat flow the observed quasi rigid behaviour of surface plates in the course of their relative motion is a consequence of the high viscosity which obtains in the cold near surface region by assigning a particular constant velocity as the upper boundary condition in a numerical model of mantle convection we have investigated the effect of lithospheric rigidity on the variation of oceanic bathymetry and heat flow as a function of ocean floor age predicted variations of both bathymetry and heat flow at the surface of mantle wide convection cells which are partially heated from within exhibit the same qualitative behaviour as data compiled for the major ocean basins the bathymetry varies with distance x from the spreading centre initially as x sup 1 2 but subsequently flattens with respect to an x sup 1 2 reference curve whereas the heat flow closely follows an x sup 1 2 decay over most of the convection cell consequently the viability of the mechanism for sea floor flattening proposed by jarvis and peltier 1980 is increased when a constant surface velocity characteristic of rigid plates is incorporated in the model this model successfully predicts both that the bathymetry should flatten and that the heat flow should not

energetics and sedimentary processes in the columbia river estuary abstract the columbia river estuary is an energetic sand bedded system mixed tides with a spring tidal range of about 3 6m and a large freshwater discharge produce bottom shear stresses capable of transporting nearly all the sizes of sediment present in the estuary as a result the morphology of the estuary is closely related to the fluxes of tidal and fluvial energy to investigate the relationship between energetics and sedimentary processes an energy budget based on a one dimensional harmonic tidal model was developed for the estuary tidal river system and has been compared with the results of geological studies of the bedforms large scale morphology sediment distribution and suspended sediment field the pattern of energy fluxes predicted by the model suggests the division of the system into three regimes a tidally dominated lower estuary a mid estuary energy flux divergence efd minimum region and a fluvially controlled tidal fluvial reach model results also show that non linear interactions between the tidal and fluvial flows are responsible for the suppression of the tides in the tidal fluvial reach during periods of high flow finally changes in tidal amplitude at the mouth result in a less than proportionate response in tidal elevations at points inside the estuary because the cubic dependence of dissipation on tidal amplitude damps system responses to such changes many features of the observed sediment transport patterns and sedimentary environments can be related to the energy budget most of the medium to coarse sands entering the system from the river are permanently retained within the efd minimum much of this deposition takes place upstream of the limits of salinity intrusion and is not therefore related to baroclinic circulatory effects most of the fine sands and the silts and clays entering the system are not permanently retained some of the silts and clays are however temporarily retained in a turbidity maximum whose mean position is near the lower end of the efd minimum this position is dictated by the inability of salinity intrusion to extend up the fluvial potential energy gradient although some aspects of sedimentary processes cannot readily be related to the energy budget our model provides a valuable conceptual approach to the dominant long term sedimentary processes of sand transport and deposition in the columbia river estuary that structure the observed sedimentary environments biological processes are in turn strongly influenced by these geological patterns

archean komatiites and geotherms solution to an apparent contradiction the high liquidus temperatures of archean komatiites and the low geothermal gradients in archean continents appear to give contradictory indications of temperatures in the archean mantle we resolve this contradiction in terms of an idealized model of convection in the archean mantle we suggest that temperatures in the archean mantle were no more than 100 c higher than at present that komatiites originate from the hot lower boundary layer of a non steady convection cell and that archean continents were formed near subduction zones where near surface temperature gradients were lower than elsewhere in the convecting mantle

low wavenumber signatures of time dependent mantle convection abstract spectral analysis is applied to idealized two dimensional numerical models of mantle convection examining the spectral signature of known model temperature fields represents the forward problem corresponding to the inverse problem of inferring the unknown temperature field in the mantle from the spectral components of lateral heterogeneity obtained by seismic tomography previous two dimensional fourier analyses of steady convection cells indicate that with current levels of resolution tomographic techniques have a much better chance of detecting horizontal boundary layers than vertical plumes from an examination of the shape of the spectral envelopes of the power spectrum of lateral heterogeneity at various depths in steady convection cells it has been suggested that evidence for thermal boundary layers in the mantle may be contained in the published spectral amplitudes of lateral heterogeneity this possibility is expanded on in the present study here we examine the effects on the spectral signature of lateral heterogeneity of averaging the model temperature field over vertical intervals of a few hundred kilometers of large amounts of internal heating of time dependent flow and of aspect ratios greater than unity the vertical averaging is an attempt to mimic the finite vertical resolution of tomographic inversions the various complications considered here are found to render the distinction between the spectra of lateral heterogeneity inside and outside of thermal boundary layers less clear than in the simpler models considered previously nevertheless the boundary layer spectra continue to contain significant information on the dominant scale of the model flow solutions

numerical modelling of convection interacting with a melting and solidification front application to the thermal evolution of the basal magma ocean abstract melting and solidification are fundamental to geodynamical processes like inner core growth magma chamber dynamics and ice and lava lake evolution very often the thermal history of these systems is controlled by convective motions in the melt computing the evolution of convection with a solid liquid phase change requires specific numerical methods to track the phase boundary and resolve the heat transfer within and between the two separate phases here we present two classes of method to model the phase transition coupled with convection the first referred to as the moving boundary method uses the finite element method and treats the liquid and the solid as two distinct grid domains in the second approach based on the enthalpy method the governing equations are solved on a regular rectangular grid with the finite volume method in this case the solid and the liquid are regarded as one domain in which the phase change is incorporated implicitly by imposing the liquid fraction f l as a function of temperature and a viscosity that varies strongly with f l we subject the two modelling frameworks to thorough evaluation by performing benchmarks in order to ascertain their range of applicability with these tools we perform a systematic study to infer heat transfer characteristics of a solidifying convecting layer parametrized relations are then used to estimate the super isentropic temperature difference maintained across a basal magma ocean bmo labrosse et al 2007 which happens to be minute 0 1 k implying that the earth s core must cool at the same pace as the bmo

mantle convection as a boundary layer phenomenon summary the boundary layer nature of vigorous thermal convection is explored using high resolution numerical solutions to the governing hydrodynamic equations these solutions are obtained for a series of idealized models of the earth s mantle in which the viscosity is assumed to be constant a detailed analysis of the local energy balance within the horizontal and vertical thermal boundary layers is presented in terms of which a test of the fundamental assumptions of boundary layer theory is provided the results of this test have important geophysical consequences since the asymptotic predictions of boundary layer theory have been employed extensively in the context of thermal history modelling although boundary layer theory closely predicts the correct power law behaviour of various quantities it does not determine their absolute values accurately vertical advection is shown to play an important role in the energy balance within horizontal boundary layers at all rayleigh numbers horizontal and vertical advection dominate the energy balance within vertical plumes while horizontal diffusion plays a very minor role when heating is partially from within the fluid vertical advection into the upper thermal boundary layer can produce significant departures in the thermal structure from that found when heating is entirely from below for a free upper boundary this results in a relative flattening of the variations of surface topography and heat flow across the convection cells for a constant velocity upper boundary similar to plate motion the bathymetry flattens but the heat flow does not this result agrees with marine observations rigidity of the thermal boundary layer below the upper surface is not included explicitly in the model and it is not known whether the inclusion of this feature in future models would significantly alter the topographic expression if not the observed departure of the oceanic bathymetry from a 6 dependence at old ocean floor ages could be attributed to a small amount of internal heating in a mantle wide convective circulation

geoid anomalies and dynamic topography from convection in cylindrical geometry applications to mantle plumes on earth and venus summary a variety of evidence suggests that at least some hotspots are formed by quasi cylindrical mantle plumes upwelling from deep in the mantle we model such plumes in cylindrical axisymmetric geometry with depth dependent newtonian viscosity cylindrical and sheet like cartesian upwellings have significantly different geoid and topography signatures however rayleigh number nusselt number systematics in the two geometries are quite similar the geoid anomaly and topographic uplift over a plume are insensitive to the viscosity of the surface layer provided that it is at least 1000 times the interior viscosity increasing the rayleigh number or including a low viscosity asthenosphere decreases the geoid anomaly and the topographic uplift associated with an upwelling plume increasing the aspect ratio increases both the geoid anomaly and the topographic uplift of a plume the nusselt number is a weak function of the aspect ratio with its maximum value occurring at an aspect ratio of slightly less than 1

dynamics of titan s high pressure ice layer abstract the presence of 40ar in titan s atmosphere and the replenishment of methane argue for the exchange between the interior and the atmosphere these observations triggered the present study that aims to determine the conditions under which the high pressure hp ice layer likely present between the deep ocean and the silicate core poses a barrier for the exchange of volatiles we model heat and water transport through this convecting hp ice layer using a two phase numerical model of solid ice liquid water mixture we observe that for a large range of heat fluxes from the silicate core and hp ice layer thicknesses a few percent of liquid water forms at the interface with the silicates liquid water being less dense than the hp ice it creates additional buoyancy thus facilitating the transport of volatiles towards the ocean our results also show that convection is characterized by the presence of hot and the absence of cold plumes we derive a scaling law that describes the dependence of a critical heat flux for the onset of melting at the silicates interface on the thickness of the hp ice layer and the ice viscosity we also study the processes at the interface with the base of the ocean where a few tens of kilometers thick layer of temperate partially molten ice is present we find a scaling law for its thickness that depends mainly on the ice viscosity and the density difference between the ice and water water from this partially molten temperate layer flows into the ocean thus completing the connection with the silicate core the water flux depends primarily on the amount of heat supplied from the silicates future evolution models that will use the scaling laws derived in this study will place bounds on the timing of these exchange processes using cassini data and reasonable values of hp ice viscosity and silicate heat flux we predict melting at the silicates hp ice interface at present time

experimental pictures of deformation patterns in a possible model of the earth s interior abstract geophysical calculations and observations of the earth s properties have previously established that the motion of tectonic plates and associated geological hazards are driven by extensive rock displacement in the mantle of our planet the progressive deformation of mantle rock by solid state convection so far only studied in two dimensional numerical computer simulations can now also be studied three dimensionally in a real fluid two billion years of structural developments in the earth s upper mantle can be visualized within one hour of flow in the laboratory assuming one particular mode of upper mantle circulation

small scale convective instability and upper mantle viscosity under california thermal calculations and convection analysis constrained by seismic tomography results suggest that a small scale convective instability developed in the upper 200 kilometers of the mantle under california after the upwelling and cooling of asthenosphere into the slab window associated with the formation of the san andreas transform boundary the upper bound for the upper mantle viscosity in the slab window 5 x 10 19 pascal seconds is similar to independent estimates for the asthenosphere beneath young oceanic and tectonically active continental regions these model calculations suggest that many tectonically active continental regions characterized by low upper mantle seismic velocities may be affected by time dependent small scale convection that can generate localized areas of uplift and subsidence

the long wavelength component of mantle convection abstract theoretical models of mantle convection predict that the major temperature fluctuations within the mantle are confined to narrow horizontal boundary layers and vertical plumes these fluctuations result in heterogeneities in seismic body wave velocities which could in principle be detected by seismic tomographic techniques however recent tomographic images of the spatial variations of temperature in the mantle are considerably out of focus in that only the longest wavelength components can be resolved to assess this partial recovery of the total tomographic image theoretical temperature fields have been generated with a numerical model of high rayleigh number mantle convection and then fourier analysed in two dimensions upon re synthesizing the model temperature fields the fourier series expansions were truncated at various levels of resolution the truncated expansions containing only the long wavelength components of the model temperature fields are compared to both the complete field and the tomographic images of the mantle at the current level of resolution it seems unlikely that seismic tomography could distinguish between layered and whole mantle convection estimates based on current tomographic data of long wavelength fluctuations of temperature and surface topography are predicted in the case of whole mantle convection to represent approximately 10 of the total temperature variation and approximately 50 of the total topographic relief thus topography at the core mantle boundary may be more accurately inferred from seismic tomography than may the characteristic lateral temperature fluctuation which drives the convective circulation

dynamics of mantle thermals with constant buoyancy or anomalous internal heating abstract two closely related problems are discussed i first apply to the earth s mantle results of recent laboratory and theoretical modelling 1 of the fluid dynamics of the smallest scales of motion in thermal convection at high rayleigh number in the mantle instabilities on a bottom boundary layer are likely to give rise to detached blobs called thermals these are predicted to be approximately spherical masses of buoyant material which enlarge by entrainment of surrounding mantle as they rise thermals are most likely to have initial diameters of about 200 km and to rise through the full depth of an ideal uniform mantle in less than 10 8 years on arrival at the base of the lithosphere their diameters may be up to 500 km they still carry almost all of the heat with which they began and they induce temperature anomalies sufficient to cause partial melting predicted length and time scales indicate that this highly unsteady form of convection may be a plausible alternative to steady deep mantle plumes as a cause of ocean island volcanism second there is the possibility that localised regions of anomalous internal heating associated with chemical heterogeneities may lead to small scale convection as plumes or thermals an analysis is presented for thermals in which all buoyancy is generated by internal heating these thermals pass through a maximum temperature as a result of assimilation of cooler surrounding mantle using as an upper limit for anomalous internal heating that due to increased concentrations of radioactive elements in subducted oceanic basalts and sediments it is shown that internal heating is of little consequence to the dynamics of chemical heterogeneities

speculations on the thermal and tectonic history of the earth summary the connection between the earth s thermal history and convection in the mantle is exploited to elucidate the early evolution of the earth it appears probable that convection extending over almost all of the mantle has dominated vertical heat transport throughout the whole of the earth s history only in boundary layers at the surface and at a depth of 650 700 km is conduction likely to be important the resulting evolution appears to be consistent with geological observations on early precambrian rocks various arguments are put forward in favour of two horizontal scales of convective flow in the mantle at depths less than 650 km the large scale flow is related to the motion of major plates and must be ordered over distances of more than 5000 km its evolution and energetics are discussed and there are no obvious problems in maintaining the proposed convective motions small scale flow with an extent of the order of 500 km appears necessary both to explain the heat flow through older parts of the earth s surface and to reconcile the geophysical observations with the results of numerical experiments though the existence of the small scale flow is at present speculative various tests of its presence are proposed

large aspect ratio convection cells in the upper mantle summary numerical experiments which model a possible large scale flow in the upper mantle are described the lithospheric plates are represented by a constant velocity boundary condition on the upper surface of a twodimensional convecting layer the transition from small scale multiple cell flow to single cell flow is described in terms of the rayleigh number and the peclet number the mean flow field is successfully compared with a simple parameterized model in which flow is driven by the moving upper boundary and by a mean horizontal temperature gradient which must be present in any large aspect ratio cell bathymetry and gravity profiles are calculated for the long cells and it is shown that in the absence of a low viscosity zone a realistic variation of bathymetry and gravity is only obtained if the flow in the cell is being driven predominantly by the horizontal temperature gradient rather than the moving plate mean horizontal temperature gradients in the upper mantle of 0 025 c km are suggested beneath a fast moving plate such as the pacific transverse boundary layer instabilities in the large cells are shown to be governed by a critical boundary layer rayleigh number which linearly increases with the boundary layer pcclet number

amplitudes of late pennsylvanian glacioeustasy the late paleozoic is well documented as a time of significant continental glaciation but the extents of the glaciation and attendant glacioeustasy are not well constrained because precise amplitudes of eustasy are difficult to extract from the stratigraphic record in this paper we use preserved relief on ancient subaerial exposure surfaces of large algal bioherms to demonstrate directly that late pennsylvanian glacioeustasy reached minimum amplitudes of 80 m and probably exceeded 100 m upper paleozoic algal bioherms accreted predominantly during sea level falls but also during sea level rises and highstands and were capable of remarkably rapid growth rates eustatic amplitudes in excess of 100 m approach amounts documented for the pleistocene and place constraints on models for gondwanan ice volume climate dynamics and potential character and magnitude of glacioclimatic fluctuations

formation of foreland rifts abstract the lower permian rotliegendes troughs in central europe are directly comparable to the upper cenozoic basin and range province in the western united states in most tectonostratigraphic aspects these intracratonic rifts were formed by extensional tectonics in the foreland area of major cordilleran type orogenies beginning 20 25 ma after mountain building the main characteristics of the basins include crustal thinning from 40 45 to 30 km change from calc alkaline magmatic arc suites to basaltic or bimodal basalt rhyolite igneous suites normal to listric block faulting high heat flow and hydrothermal activity regional uplift and immature clastic and evaporite sediments rapidly deposited within the subsiding rifts in a semi desert environment the genetic model proposed for these foreland rifts assumes that an extensional force under the continental foreland results from mantle material being drawn back towards the subducting plate by viscous drag this force is counteracted by a compressional force produced at the trench by subducting a young relatively buoyant plate until subduction is ended at which time the plate breaks off and continues to sink tension in the foreland possibly produced by the sinking slab continues for some time later causing stretching and thinning of the continental crust extensional tectonics and higher regional heat flow and uplift the decrease in overburden pressure and higher heat flow induces partial melting in the lower lithosphere and bimodal volcanism hotter asthenosphere rising up behind the trailing edge of the sinking plate causes thermal expansion and further uplift the heat flow anomaly rapidly decays by hydrothermal activity within 15 20 ma after last volcanism block faulting and rift basin infilling follow the main period of volcanism and continue until the sinking slab is decoupled from the overriding plate and no longer influences the continental foreland

lateral heterogeneity in the convecting mantle thermally induced lateral heterogeneity in the earth s mantle has been studied in terms of idealized two dimensional constant viscosity numerical models of mantle convection steady model solutions of the temperature and velocity fields were analyzed with respect to both the spatial and spectral signatures of their inherent lateral heterogeneity by examining the changes in spectral signature produced by varying the rayleigh number the degree of internal heating and the depth in the convection cell it has been possible to identify those features of the spectra of lateral heterogeneity which are most indicative of these variations particular attention has been paid to the spectral characteristics of thermal boundary layers within the convecting layer the spectral analysis of known temperature fields represents the forward problem corresponding to the inverse problem of inferring the thermal structure of the mantle from the spectral components of its lateral heterogeneity as determined by seismic tomography the forward problem presented here illustrates a new approach in convection modeling which may ultimately provide important diagnostic criteria for the interpretation of recent tomographic data

high spatial resolution models of time dependent layered mantle convection summary the degree of geochemical mixing that occurred during the mantle s early evolution is dependent on the history of convective motions in the mantle we present time dependent tnodels of mantle convection that show for certain conditions in the newly formed earth s mantle a transient phase of layered convection results during which geochemical differences between the upper and lower mantle could have been preserved the basic model assumes that the initial temperature profile for subsolidus mantle convection is strongly superadiabatic the mantle is represented by a constant viscosity boussinesq fluid in a 2 d cartesian box heated from the bottom the models have considerably greater spatial resolution 90 x 90 uniformly spaced grid points than our previous work parameters that have been vaned include the rayleigh number ra lo5 lo8 the aspect ratio 1 4 9 the buoyancy of the initial temperature profde and the maximum temperatures allowed for subsolidus convection the models support our previous assertion that there is a critical value of ra above which superadiabatic initial conditions result in a burst of convection that inverts the temperature gradient in the bulk of the mantle sufficiently to temporarily prohibit a single layer of convection instead two layers result this critical value is about ra lo6 for a mantle with the maximum amount of buoyancy for less buoyant initial conditions this critical value is somewhat higher the layering lasts on the order of lo9 yr for ra lo6 10 considering that ra for the early mantle was certainly much larger than lo6 this suggests that a transient phase of multiple layering may have prevented vigorous convection from geochemically homogenizing the early mantle however if a less buoyant initial condition prevailed or one where supersolidus convection controls the maximum temperature of the upper mantle the transient layering could not have taken place

secular cooling of the earth as a source of intraplate stress abstract as a result of secular cooling and contraction of the earth an increasingly extensional horizontal stress is imposed on the central portions of the plates while the rate of increase of this stress is small in most situations it is enhanced for large plates and during any episodes of accelerated global cooling this source of stress may have contributed to the rifting and breakup of long lived supercontinental plates

two phase natural convection implications for crystal settling in magma chambers abstract crystal settling is no longer thought to be the most likely mechanism operating in the formation of layered igneous intrusions the main objection raised against its feasibility is the prediction that crystal settling velocities are likely to be orders of magnitude smaller than convective velocities and that vigorous convection would therefore keep crystals well mixed within the interior of a magma chamber in previous theoretical and numerical investigations of crystal settling no account has been taken of compositional buoyancy gradients created by the interpenetration of crystals and melt this paper includes these effects the results of two dimensional numerical simulations show that parameter regimes exist in which effective crystal settling occurs in infinite prandtl number thermally convecting two phase fluids even for small ratios of settling to convective velocity

convection with internal heat sources and thermal turbulence in the earth s mantle summary the influence of internal heat sources on mantle convection is investigated using numerical calculations of 2 d thermal convection in an infinite prandtl number incompressible fluid the geometry is a cylindrical annulus with inner and outer radii in proportion to the whole mantle time dependent calculations are made starting from random initial conditions with rayleigh numbers rat based on boundary temperature difference and ra based on internal heat production in the range 1dsrats lo7 and osra 524rat at fixed rat increasing ra results in transitions in flow structure from steady cells to a pattern of stationary cells with time variable amplitude and finally to thermally turbulent convection with a non stationary cell count for rat 10 and ra rat approximately the travelling plumes disrupt the large scale circulation producing turbulent convection at ra lo7 the flow is fully developed thermal turbulence and for ra 0 consists of a rapidly fluctuating irregular flow driven by transient rising and sinking sheets of buoyant fluid large fluctuations in total kinetic energy occur in this regime with periodicities ranging from 40 to 1400myr the transition to thermal turbulence occurs in these calculations at rayleigh numbers well below the value estimated for subsolidus convection in the mantle suggesting thermally turbulent convection may occur in the mantle a consequence of internal heat sources thermal turbulence offers an explanation for long term fluctuations in the rate of subduction sea floor spreading and global volcanic activity