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A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number: Effects of depth-dependent viscosity, heating mode, and an endothermic phase change
Abstract
Mantle convection is influenced simultaneously by a number of physical effects: brittle failure in the surface plates, strongly variable viscosity, mineral phase changes, and both internal heating (radioactivity) and bottom heating from the core. Here we present a systematic study of three potentially important effects: depth-dependent viscosity, an endothermic phase change, and bottom versus internal heating. We model three-dimensional spherical convection at Rayleigh Ra=108 thus approaching the dynamical regime of the mantle. An isoviscous, internally heated reference model displays point-like downwellings from the cold upper boundary layer, a blue spectrum of thermal heterogeneity, and small but rapid time variations in flow diagnostics. A modest factor 30 increase in lower mantle viscosity results in a planform dominated by long, linear downwellings, a red spectrum, and great temporal stability. Bottom heating has the predictable effect of adding a thermal boundary layer at the base of the mantle. We use a Clapeyron slope of gamma=-4MPa°K-1 for the 670 km phase transition, resulting in a phase buoyancy parameter of P=-0.112. This phase change causes upwellings and downwellings to pause in the transition zone but has little influence on the inherent time dependence of flow and only a modest reddening effect on the heterogeneity spectrum. Larger values of P result in stronger effects, but our choice of P is likely already too large to be representative of the mantle transition zone. Combinations of all three effects are remarkably predictable in terms of the single-effect models, and the effect of depth-dependent viscosity is found to be dominant.
... We solve the forward mantle convection equations that comprise the principles of conservation of mass, momentum and energy. To this end, we model three-dimensional (3-D) time-dependent incompressible mantle convection via the Boussinesq approximation (Chandrasekhar, 1961) and solve the equations using the finite element TERRA code (Bunge et al. 1996(Bunge et al. , 1997) on a cluster optimized for geophysics-capacity simulations (Oeser et al., 2006). The equations are presented below: ...
... In all our experiments, the isoviscous internal heating case, which represents the base scenario, provided the best results for velocity assimilation. This class of models has a very simple thermal profile with only an upper thermal boundary layer, a flow pattern dominated by active downwellings (Bunge et al., 1997) and structures rising and sinking at relatively uniform velocity. The absence of a lower thermal boundary layer in this class of models means that there are no actively rising plumes whose locations would be unconstrained because, by nature of the geophysical problem, velocity assimilation cannot happen at the CMB. ...
... In a compressible mantle, the flow slows down with depth but has faster velocities close to the surface in comparison to an incompressible mantle (Bunge et al., 1997). This is necessary in order to maintain constant mass flux (Panasyuk et al., 1996). ...
The ability to construct time-trajectories of mantle flow is crucial to move from studies of instantaneous to time-dependent Earth models and to exploit geologic constraints for mantle convection modelling. But mantle convection is chaotic and subject to the butterfly effect: the trajectories of two identical mantle convection models initialized with slightly different temperature fields diverge exponentially in time until they become uncorrelated. Because one may use seismic inferences about the mantle state as a starting or terminal condition to project mantle flow forward or backward in time, and because the seismic inference is invariably subject to uncertainties, this seemingly would rule out any construction of robust mantle flow trajectories. Here we build upon earlier work which showed that assimilation of the horizontal component of the surface velocity field from a known reference model allows one to overcome the butterfly effect and to construct robust mantle flow trajectories, regardless of the choice of the initial state perturbation. To this end, we use high resolution 3-D spherical mantle convection models in four end-member configurations: an isoviscous purely internally heated model, an isoviscous purely bottom heated model, a model with a radial increase in viscosity along with pure internal heating as well as a model that combines the effects of radial viscosity increase, internal and bottom heating. In order to capture the impact of seismic filtering, we perturb the initial temperature fields of these end-member models through either radial or horizontal smoothing of the temperature field or the application of the tomographic filter of seismic model S20RTS. We assess the quality of the constructed model trajectories via a number of statistical measures as well as comparisons of their dynamic topography histories. The latter is an essential step since mantle flow cannot be directly observed but has to be inferred via its surface manifestations. Importantly, linking mantle flow to surface observations yields patterns representable on a latitude-longitude grid similar to meteorological observables such as precipitation. This invites the application of meteorological quality metrics, such as the power ratio and Taylor diagram, to assess the quality of mantle flow trajectories. We introduce these metrics for the first time in the context of mantle convection and demonstrate their viability based on the compact manner in which they summarize model performance.
... Christensen & Yuen 1984, 1985Christensen 1995). Material flow in the TZ will further be influenced by the collocated increase in viscosity (Hager & Richards 1989;Bunge et al. , 1997Busse et al. 2006;Paulson & Richards 2009;Schaber et al. 2009;Leng & Gurnis 2012). Furthermore, the pyroxene-normative phase transitions add to the complexity of flow and heterogeneity in the TZ, in particular those of garnet that may also contribute to discontinuity topography (e.g. ...
... Numerous studies concerned with mantle seismic structure, mineral physics or geodynamic modelling have contributed to providing constraints on TZ structure as well as a better understanding of its role in mantle flow (Morgan & Shearer 1993;Tackley et al. 1993Tackley et al. , 1994Christensen 1995;Bunge et al. 1997;Weidner & Wang 1998, 2000Fukao et al. 2001;Nolet et al. 2006; Thomas & Billen 2009;Ishii et al. 2011;Saki et al. 2015;Reiss et al. 2017Reiss et al. , 2018Ishii et al. 2018). Experimental studies including Ishii et al. (2011Ishii et al. ( , 2018Ishii et al. ( , 2019 and Hirose (2002) have introduced constraints on the exact P-T conditions of the major mineral phase transitions. ...
... Since, however, a link to mantle thermal structure has not been made in these studies, they do not allow to investigate thermally induced effects on mineral phase transformations or the dynamic impact of the TZ. Geodynamic studies such as Tackley et al. (1993Tackley et al. ( , 1994 and Bunge et al. (1997), on the other hand, explored the role of the TZ on flow dynamics by implementing phase transition effects in models of mantle convection. They specifically focused on the endothermic transition at the base of the upper mantle and the associated buoyancy effects. ...
The mantle transition zone (TZ) is expected to influence vertical mass flow between upper and lower mantle as it hosts a complex set of mineral phase transitions and an increase in viscosity with depth. Still, neither its seismic structure nor its dynamic effects have conclusively been constrained. The seismic discontinuities at around 410 and 660 km depth (‘410’ and ‘660’) are classically associated with phase transitions between olivine polymorphs, the pressure of which is modulated by lateral temperature variations. Resulting discontinuity topography is seismically visible and can thus potentially provide insight on temperature and phase composition at depth. Besides the olivine phase changes, the disassociation of garnet may additionally impact the 660 at higher temperatures. However, the volume of material affected by this garnet transition and its dynamic implications have not yet been quantified. This study presents hypothetical realizations of TZ seismic structure and major discontinuities based on the temperature field of a published 3-D mantle circulation model for a range of relevant mineralogies, including pyrolite and mechanical mixtures (MM). Systematic analysis of these models provides a framework for dynamically informed interpretations of seismic observations and gives insights into the potential dynamic behaviour of the TZ. Using our geodynamic-mineralogical approach we can identify which phase transitions induce specific topographic features of 410 and 660 and quantify their relative impact. Areal proportions of the garnet transition at the 660 are ∼3 and ∼1 percent for pyrolite and MM, respectively. This proportion could be significantly higher (up to ∼39 percent) in a hotter mantle for pyrolite, but remains low (< 2 percent) for MM. In pyrolite, both slabs and plumes are found to depress the 660 —with average deflections of 14 and 6 km, respectively— due to the influence of garnet at high temperatures indicating its complex dynamic effects on mantle upwellings. Pronounced differences in model characteristics for pyrolite and MM, particularly their relative garnet proportions and associated topography features, could serve to discriminate between the two scenarios in Earth.
... The approximation was initially proposed for situations with large Reynolds numbers such as the atmosphere (Ogura and Phillips, 1962), the liquid core (Braginsky and Roberts, 1995) or the stars (Lantz and Fan, 1999), and was later applied to low Reynolds number situations. The rich bibliography of numerical studies of convection applied to planetary interiors is based on Boussinesq models (e.g., Blankenbach et al., 1989;Busse et al., 1994;Bunge et al., 1997;Parmentier and Sotin, 2000;Choblet et al., 2007;Zhong et al., 2008) or Anelastic models (e.g., Jarvis and McKenzie, 1980;Glatzmaier, 1988;Bercovici et al., 1989Bercovici et al., , 1992Tackley, 2008;Rolf et al., 2012;Kameyama and Yamamoto, 2018). A comprehensive presentation of the fluid dynamic equations and their approximations when they are applied to planetary interiors is found in Schubert et al. (2001). ...
... The value of n (also sometimes referred to as K 0 ) is ≈ 3−4 for most mantle materials (Stixrude and Lithgow-Bertelloni, 2005). Equation (4) has been used implicitly in various models of mantle convection (e.g., Glatzmaier, 1988;Bercovici et al., 1989Bercovici et al., , 1992Bunge et al., 1997). It can easily be used to derive any thermodynamic property like the thermal expansion coefficient α(P, T ) or the incompressibility K T (P, T ). ...
... A choice of an appropriate EoS allows all the terms needed by the formalism to be computed and the Murnaghan equation of state with a Grüneisen parameter varying as the inverse of density seems a flexible and accurate choice. Although we are certainly not the first ones to start with a Murnaghan EoS (e.g., Glatzmaier, 1988;Bercovici et al., 1989Bercovici et al., , 1992Bunge et al., 1997), all previous authors seem to have added additional approximations (such as constant parameters) that were not necessary. Various ingredients of planetary dynamics are not considered in our 2D simulations, such as 3D geometry, sphericity, depth dependence of gravity, viscosity or conductivity but should be easily implemented (although the computation time would largely increase). ...
The numerical simulations of convection inside the mantle of the Earth or of terrestrial planets have been based on approximate equations of fluid dynamics. A common approximation is the neglect of the inertia term which is certainly reasonable as the Reynolds number of silicate mantles, or their inverse Prandtl number, are infinitesimally small. However various other simplifications are made which we discuss in this paper. The crudest approximation that can be done is the Boussinesq approximation (BA) where the various parameters are constant and the variations of density are only included in the buoyancy term and assumed to be proportional to temperature with a constant thermal expansivity. The variations of density with pressure and the related physical consequences (mostly the presence of an adiabatic temperature gradient and of dissipation) are usually accounted for by using an anelastic approximation (AA) initially developed for astrophysical and atmospheric situations. The BA and AA cases provide simplified but self-consistent systems of differential equations. Intermediate approximations are also common in the geophysical literature although they are invariably associated with theoretical inconsistencies (non conservation of total energy, non conservation of statistically steady state heat flow with depth, momentum and entropy equations implying inconsistent dissipations). We show that, in the infinite Prandtl number case, solving the fully compressible (FC) equations of convection with a realistic equation of state (EoS) is however not much more difficult or numerically challenging than solving the approximate cases. We compare various statistical properties of the Boussinesq, AA and FC simulations in 2D simulations. We point to an inconsistency of the AA approximation when the two heat capacities are assumed constant. We suggest that at high Rayleigh number, the profile of dissipation in a convective mantle can be directly related to the surface heat flux. Our results are mostly discussed in the framework of mantle convection but the EoS we used is flexible enough to be applied for convection in icy planets or in the inner core.
... Our models build on two previously available codes, one for simulating mantle convection (TERRA), the other for simulating lithosphere dynamics (SHELLS). TERRA is a global, spherical finite element code developed and on which this article is based on are available in Baumgardner (1985) and Bunge and Baumgardner (1995), and further advanced by Bunge et al. (1997); Davies et al. (2013); Stegman et al. (2003); Yang (1997), among others. TERRA solves the classical conservation equations (energy, momentum, and mass) at infinite Prandtl number within a spherical shell domain. ...
... We begin by generating global temperature, pressure and velocity fields for the mantle using a modified and benchmarked version of the spherical mantle convection code TERRA. As in the simulations presented in Davies et al. (2012), (2015), calculations are performed on a numerical mesh with ∼80 million discrete nodal points, thus providing the resolution necessary to explore mantle flow at Earth-like convective vigor: Models achieve an internally heated Rayleigh number, based upon reference values, of 5 ⋅ 10 8 , which is similar to estimates of the mantle's Rayleigh number (e.g., Bunge et al., 1997). Our models incorporate compressibility, in the form of the anelastic liquid approximation, with radial reference values represented through a Murnaghan equation of state. ...
... TERRA is available through Baumgardner (1985); Bunge and Baumgardner (1995); Bunge et al. (1997); Davies et al. (2013); Davies et al. (2015); Stegman et al. (2003); Yang (1997) and SHELLS is available through Bird (1988); Kong and Bird (1995). All input files necessary to run the models presented as part of this study can be found on the following Zenodo online repository: https://zenodo.org/record/3937316. ...
The separation between Australia and Antarctica occurred during the final stages of the break‐up of Pangea. Reconstructions of the rifting of the Australian plate away from Antarctica show fast spreading rates since Mid‐Eocene (45 Ma). These reconstructions can be used to understand and quantify the forces driving the Australia/Antarctica separation, and to test hypotheses on mechanisms that may be of shallow (i.e., lithosphere) or deep (i.e., mantle) origin. Analytical calculations indicate that plate‐boundary forces are highly unlikely to be a plausible candidate to explain such a separation. Thus, we use a recently developed global coupled models of mantle and lithosphere dynamics, here we show that this event, whose kinematics are reproduced in our models within the bounds of the reconstruction uncertainties, owes to a significant degree to the pressure‐driven asthenospheric Poiseuille flow associated with the mantle buoyancy field inherited from viscous circulation history throughout the Mesozoic. On the contrary, in simulations when such a buoyancy field is replaced by another one resulting from a random distribution of mantle temperature–thus not representative of Earth’s mantle circulation history–the rapid northward motion of Australia does not occur. Similarly, suppressing contemporaneous plate‐boundary processes (i.e., subduction of the Pacific ridge at the Aleutians and healing of the India‐Australia ridge) from our models does not have a noticeable effect on the Australia‐Antarctica kinematics. Thus, a pressure‐driven Poiseuille mantle flow must be considered, at least in this example and possible elsewhere, as a main driver of plate tectonics.
... The unusually low number of upwellings results from the stepwise increase in viscosity at 670 km depth. Such an increase in lower mantle viscosity reddens the thermal heterogeneity spectrum [7]. It also favors the development of sheet-like and elongated downwellings. ...
... The temperature ¢eld at 700 km depth of these experiments is shown in Fig. 3a^d. The free-slip layered viscosity case (Fig. 3c) features the ability of viscosity increase with depth to promote elongated cold structures instead of pointlike downwellings characterizing the pure internally heated isoviscous case (Fig. 3a) [7,23]. This contrasts with the similarity in the convection planform depicted by plate cases (isoviscous in Fig. 3b and layered viscosity in Fig. 3d). ...
... A greater viscosity step should reduce the mean mantle temperature allowing higher CMB temperatures, but also should increase the heat £ux from the core up to unrealistic values. On the other hand, the endothermic phase change responsible for the 650 km depth seismic discontinuity does not appear to be able to stratify the convection in a two layer mode with a signi¢cant thermal boundary layer at the phase change [7,39]. A more promising way to conciliate the paradox existing between the sublithospheric temperature, the CMB temperature and the small heat £ux at the core surface, perhaps lies in the existence of geochemical strati¢cation of the mantle as proposed by Davaille [40] and Kellogg et al. [41]. ...
A simple three-dimensional spherical model of mantle convection, where plates are taken into account in the top boundary condition, allows to investigate the plate tectonics^mantle convection coupling in a self-consistent way. Avoiding the strong difficulties inherent in the numerical treatment of rheology, the plate condition appears efficient in reproducing the Earth-like features as subduction, mid-oceanic ridges and hotspots. Whereas the free-slip condition leads to a classical polygonal cell pattern with cylindrical hot plumes surrounded by downwellings, the plate condition favors the development of strong linear downwellings associated to passive diverging zones along plate boundaries. These cold currents, very similar to subductions, act the main role in mantle convection: they drive the whole circulation. In that context, hot plumes remain almost independent, except if on the long term, cold material spreading at the core surface induces a slight migration, below a few mm/yr, of their surface impingement. The main result is that plate tectonics appear to be more than a simple mode of organization of the surface movements, it is the essence of the Earth mantle dynamics. ß
... The first approaches to mantle convection were through 2-D models. However, nowadays the mantle convection is also modeled by means of 3-D models [5]. ...
... One of the most argued theories is the one of two layers of convection due to the seismic discontinuity of Repetti which is approximately at 660 km of depth, [5]. The existence of matter transport through this discontinuity was verified with the development of the seismic tomography, [5]. ...
... One of the most argued theories is the one of two layers of convection due to the seismic discontinuity of Repetti which is approximately at 660 km of depth, [5]. The existence of matter transport through this discontinuity was verified with the development of the seismic tomography, [5]. However, it is known that the transition zone marks a change in the convection of the mantle, which makes possible to think that at some early stage of Earth evolution there was a convection in two layers due to convection forces of the Earth [3,4,6]. ...
As it is well known both atmospheric and mantle convection are very complex phenomena. The dynamical description of these processes is a very difficult task involving complicated 2-D or 3-D mathematical models. However, a first approximation to these phenomena can be by means of simplified thermodynamic models where the restriction imposed by the laws of thermodynamics play an important role. An example of this approach is the model proposed by Gordon and Zarmi in 1989 to emulate the convective cells of the atmospheric air by using finite-time thermodynamics (FTT). In the present article we use the FTT Gordon-Zarmi model to coarsely describe the convection in the Earth’s mantle. Our results permit the existence of two layers of convective cells along the mantle. Besides the model reasonably reproduce the temperatures of the main discontinuities in the mantle, such as the 410 km-discontinuity, the Repetti transition zone and the so-called D-Layer.
... This conundrum motivated mantle convection simulations that explored a range of factors that could generate long wavelength mantle convection. One factor that was found to contribute within numerical models, albeit not necessarily the only factor that could, was the presence of an upper mantle layer with viscosity lower than plates above and bulk mantle below, i.e., a model analog for the asthenosphere 87,88 . ...
... The simulations of Bunge et al. 87,88 , and subsequently others [89][90][91][92] , identified a connection between the asthenosphere and mantle dynamics but did not address how, physically, the asthenosphere could contribute to the existence of long wavelength flow in the Earth's mantle. Theoretical scaling analysis showed that flow channelization within the asthenosphere is a critical factor 93,94 . ...
The existence of a thin, weak asthenospheric layer beneath Earth’s lithospheric plates is consistent with existing geological and geophysical constraints, including Pleistocene glacio-isostatic adjustment, modeling of gravity anomalies, studies of seismic anisotropy, and post-seismic rebound. Mantle convection models suggest that a pronounced weak zone beneath the upper thermal boundary layer (lithosphere) may be essential to the plate tectonic style of convection found on Earth. The asthenosphere is likely related to partial melting and the presence of water in the sub-lithospheric mantle, further implying that the long-term evolution of the Earth may be controlled by thermal regulation and volatile recycling that maintain a geotherm that approaches the wet mantle solidus at asthenospheric depths.
... In the Rayleigh-Bénard system, these spacings are comparable to the layer thickness, but in planetary mantles several peculiarities promote convection cells with large aspect ratios. These include the strength of the lithosphere (e.g., van Heck and Tackley 2008;Yoshida 2008;Rolf et al. 2014Rolf et al. , 2018a, pressure-dependence of mantle viscosity (Bunge et al. 1997;Lenardic 2008, 2010;Höink et al. 2012;Lenardic et al. 2019) and other material properties (e.g., Hansen et al. 1993), as well as the mantle heating mode (McNamara and Zhong 2005). On Earth, this is manifested in the size of the largest tectonic plates, like the Pacific plate. ...
... Depending on the ambient temperature-pressure conditions, mantle rocks undergo a series of phase transitions that modify crystallographic structure. Both density and viscosity change, which possibly promotes separation of distinct layers (Weidner and Wang 2000) and affects mantle flow structure and radial heat transport (Tackley 1996;Bunge et al. 1997). Certain components may become abundant in specific regions of the mantle. ...
The dynamics and evolution of Venus’ mantle are of first-order relevance for the origin and modification of the tectonic and volcanic structures we observe on Venus today. Solid-state convection in the mantle induces stresses into the lithosphere and crust that drive deformation leading to tectonic signatures. Thermal coupling of the mantle with the atmosphere and the core leads to a distinct structure with substantial lateral heterogeneity, thermally and compositionally. These processes ultimately shape Venus’ tectonic regime and provide the framework to interpret surface observations made on Venus, such as gravity and topography. Tectonic and convective processes are continuously changing through geological time, largely driven by the long-term thermal and compositional evolution of Venus’ mantle. To date, no consensus has been reached on the geodynamic regime Venus’ mantle is presently in, mostly because observational data remains fragmentary. In contrast to Earth, Venus’ mantle does not support the existence of continuous plate tectonics on its surface. However, the planet’s surface signature substantially deviates from those of tectonically largely inactive bodies, such as Mars, Mercury, or the Moon. This work reviews the current state of knowledge of Venus’ mantle dynamics and evolution through time, focussing on a dynamic system perspective. Available observations to constrain the deep interior are evaluated and their insufficiency to pin down Venus’ evolutionary path is emphasised. Future missions will likely revive the discussion of these open issues and boost our current understanding by filling current data gaps; some promising avenues are discussed in this chapter.
... Studies that included only the major olivine phase transitions and used comparatively large negative values for the Clapeyron slope of the post-spinel transition in the range of −3 to −4 MPa K −1 have found that it can prevent both upwellings and downwellings from penetrating the transition, leading to intermittent layering of mantle convection and trapping some heat and mass at depth. Cold downwellings have been found to accumulate in the transition zone before causing complete overturns in large avalanche events (Christensen & Yuen 1985;Peltier & Solheim 1992;Tackley et al. 1993;Bunge et al. 1997), and plume material may be deflected laterally and be retained partially or completely at the top of the lower mantle (Davies 1995;Marquart & Schmeling 2000;Tosi & Yuen 2011). Details also depend on the rheology used in a model, trench kinematics and the specific phase transitions that are included. ...
... In the past, geodynamic modelling studies have used two main numerical methods for implementing phase transitions. In early studies (Richter 1973;Christensen & Yuen 1984, 1985Peltier & Solheim 1992;Tackley et al. 1993;Zhong & Gurnis 1994;Bunge et al. 1997), individual phase transitions were often parametrized and implemented in the form of an analytical function, the 'phase function' (e.g. Christensen & Yuen 1985), which describes the proportion of each stable phase in dependence of depth (or hydrostatic pressure). ...
Phase transitions play an important role for the style of mantle convection. While observations and theory agree that a substantial fraction of subducted slabs and rising plumes can move through the whole mantle at present day conditions, this behavior may have been different throughout Earth’s history. Higher temperatures, such as in the early Earth, cause different phase transitions to be dominant, and also reduce mantle viscosity, favoring a more layered style of convection induced by phase transitions. A period of layered mantle convection in Earth’s past would have significant implications for the secular evolution of the mantle temperature and the mixing of mantle heterogeneities. The transition from layered to whole mantle convection could lead to a period of mantle avalanches associated with a dramatic increase in magmatic activity. Consequently, it is important to accurately model the influence of phase transitions on mantle convection. However, existing numerical methods generally preclude modeling phase transitions that are only present in a particular range of pressures, temperatures or compositions, and they impose an artificial lower limit on the thickness of phase transitions. To overcome these limitations, we have developed a new numerical method that solves the energy equation for entropy instead of temperature. This technique allows for robust coupling between thermodynamic and geodynamic models and makes it possible to model realistically sharp phase transitions with a wide range of properties and dynamic effects on mantle processes. We demonstrate the utility of our method by applying it in regional and global convection models, investigating the effect of individual phase transitions in the Earth’s mantle with regard to their potential for layering flow. We find that the thickness of the phase transition has a bigger influence on the style of convection than previously thought: With all other parameters being the same, a thin phase transition can induce fully layered convection where a broad phase transition would lead to whole-mantle convection. Our application of the method to convection in the early Earth illustrates that endothermic phase transitions may have induced layering for higher mantle temperatures in the Earth’s past.
... The time-dependent mantle convection flow field is solved using the governing equations for mantle convection (Mckenzie et al., 1974) and the robust three-dimensional mantle convection code TERRA (Baumgardner, 1985;Bunge & Baumgardner, 1995;Bunge et al., 1997;Davies et al., 2013). Calculations were performed on a mesh with over 10 million grid points, giving an average grid spacing of 45 km over the whole mantle volume. ...
... The inclusion of a phase change at either or both boundaries at 410-and 660-km depth (Bunge et al., 1997;Tackley et al., 1994) all result in similar increases in the total mantle water abundance as the density cases ( Figure 4d). A phase change at 410-km depth enables the cold downwelling material to descend quicker into this region resulting in an increase in the presence of colder than average material which allows more water to be stored. ...
Earth's mantle is known to harbor water in the form of hydrous and nominally anhydrous minerals. How much water the mantle holds and whether it has remained constant through time are open questions. Previous numerical studies of the deep-water cycle have been limited to box models or 2-D calculations. Here we present for the first time results from 3-D mantle convection models. We address the evolution of the mantle's total water content by adapting a well benchmarked mantle convection code to track water, including its feedbacks on dynamics. While Earth's surface is presently covered by one ocean mass of water, our results suggest that the mantle holds approximately two ocean masses of water based on the best estimates from mineral physics. This value varies only weakly for a wide parameter space of additional complex dynamics such as viscosity laws, density controls, and phase change considerations. Our result of a mantle holding two ocean masses conforms with estimates from other branches of Earth science, suggesting that these models could be an excellent tool in understanding the spatial heterogeneity of the water found in the mantle.
... This presented a question: What allows for long wavelength convection? A potential solution came from numerical simulations that generated longwavelength convection by imposing a viscosity increase from the upper to the lower mantle (Bunge et al., 1996(Bunge et al., , 1997Hansen et al., 1993;Tackley, 1996;Zhang and Yuen, 1995;Zhong et al., 2007). The physical mechanism behind this observation was elucidated via boundary-layer theory Lenardic et al., 2006). ...
Isolating planetary feedbacks and feedback analysis are prevalent aspects of climate and Earth surface process science. An under-appreciation of internal planet feedbacks, and feedback analysis for plate tectonics research, motivates this chapter. We review feedbacks that influence the Earth’s thermal evolution and expand them to include magmatic history and planetary water budgets. The predictions from feedback models are shown to be consistent with petrological constraints on the Earth’s cooling. From there, we isolate feedbacks that connect structural elements within the mantle dynamics and plate tectonics system. The feedbacks allow for a reciprocal causality between plates, plumes, the asthenosphere, and mantle flow patterns, with each element being co-dependent on the others. The linked elements and feedbacks define plate tectonics are part of a self-sustaining flow system that can bootstrap itself into existence. Within that framework, plate tectonics involves the co-arising of critical system factors. No single factor is the cause of another. Rather, they emerge with the links between them and the generation of functional elements coincides, within relatively narrow time windows, with the co-emergence of factors that are critical for the maintenance of the elements themselves. What emerges is not a tectonic state but a process. That is, a set of feedbacks can transform the tectonics of a planet and/or maintain plate tectonics. The feedback functions are not permanent but can operate over extended time frames such that plate tectonics can remain stable. The nature of the feedbacks, and their stability, can be studied at various levels of detail but questions of origin can become ill-defined. Observational tests of a feedback framework for plate tectonics and mantle dynamics are presented, along with research paths that apply feedback methodology to solid-planet dynamics and comparative planetology.
... In this study we use the three-dimensional mantle convection code, TERRA (Baumgardner, 1985;Bunge et al., 1997; D. R. Davies et al., 2013) to solve the governing equations for mantle convection under the Boussinesq 10.1029/2022JB025610 3 of 21 approximation and assuming incompressibility (McKenzie et al., 1974). Values for common parameters can be found in Table 1. ...
The large low‐shear‐velocity provinces (LLSVPs) are thought to be thermo‐chemical in nature, with recycled oceanic crust (OC) being a contender for the source of the chemical heterogeneity. The melting process which forms OC concentrates heat producing elements (HPEs) within it which, over time, may cause any collected piles of OC to destabilize, limiting their suitability to explain LLSVPs. Despite this, most geodynamic studies which include recycling of OC consider only homogeneous heating rates. We perform a suite of spherical, three‐dimensional mantle convection simulations to investigate how buoyancy number, geochemical model and heating model affects the ability of recycled OC to accumulate at the core‐mantle boundary. Our results agree with others that only a narrow range of buoyancy numbers allow OC to form piles in the lower mantle which remain stable to present day. We demonstrate that heterogeneous radiogenic heating causes piles to destabilize more readily, reducing present day CMB coverage from 63% to 47%. Consequently, the choice of geochemical model can influence pile formation. Geochemical models which lead to high internal heating rates can cause more rapid replenishment of piles, increasing their longevity. Where piles do remain to present day, first order comparisons suggest that old (hot) OC material can produce seismic characteristics, such as Vs anomalies, similar to those of LLSVPs. Given the range of current density estimates for lower mantle mineral phases, subducted OC remains a contender for the chemical component of thermo‐chemical LLSVPs.
... Müller et al. (2018) represent the second model that created a threefold strategy of mantle-driving dynamic topography evolved through time for the passive margins, continental interior, and hinterlands. Another quantitative framework by Rubey et al. (2017) was proposed to establish a connection between convecting mantle (using the "TERRA" code of Bunge et al., 1997;Baumgardner, 1985;Wolstencroft et al., 2009;Davies and Davies, 2009;Wolstencroft and Davies, 2017;Davies et al., 2013) and basin evolution and plate tectonics. The fourth model of Flament et al. (2013) focused on eliminating the isostatic influence of lithosphere/crust, ice, sediments, and water from the observed topography, which is likely the residual topography. ...
Tectonic processes of the SE Tibet Plateau (SETP) and the northern South China Sea margin (NSCSM) strongly affected the surface drainage system for sediment transportation to deposition. It remains challenging to investigate the dominant mechanisms such as tectonics, deep dynamics, and surface processes. This paper aims to examine the influence of the SETP paleogeography on the NSCSM. Twenty-eight seismic profiles were quantified by flexurally interpolated backstripping and significantly correlated with the published paleogeography. A numerical simulation tool (Badlands) is assigned to study the integrated response of surface processes and deep dynamics. This paper inferred that drastic sedimentation changes in our models were linearly correlated mainly with erosion during the SETP uplifts, dynamic topography, and flexural isostasy. Deep dynamics and tectonic processes control basin architectures for sediment depositions, whereas the surface processes contribute isostatically in response to uplifts in the source region. We inferred that Eocene was the stage of extension, thinning of lithosphere (Te = 10 km) that is linked with the low topography (<2000 m) and erodibility (Є) of 5e-7 in the SETP. However, the increase in cumulative thickness is associated with strengthening the lithosphere (Te = 15 km) and tributary rotation with its rise in SETP topography (≥4000 m) during the Oligocene to Early Miocene. The Middle Miocene doesn't have the source-sink linkage for its high SETP topography with low deposition. But a strong connection with paleogeographic evolution in the later stage due to the rise in SETP topography and Taiwan orogeny.
... We use the three-dimensional mantle convection code, TERRA (Baumgardner, 1985;Bunge & Baumgardner, 1995;Bunge et al., 1997;Davies et al., 2013), to solve the governing equations for mantle convection. We apply the Boussinesq approximation and assume incompressibility (McKenzie et al., 1974) to give the equations for conservation of mass (Equation 2), momentum (Equation 3), energy (Equation 4), and bulk composition (Equation 5). ...
For mid‐ocean ridge basalts and ocean island basalts, measurements of Pb isotope ratios show broad linear correlations with a certain degree of scatter. In ²⁰⁷Pb/²⁰⁴Pb—²⁰⁶Pb/²⁰⁴Pb space, the best fit line defines a pseudo‐isochron age (τPb) of ∼1.9 Gyr. Previous modeling suggests a relative change in the behaviors of U and Pb between 2.25 and 2.5 Ga, resulting in net recycling of HIMU (high U/Pb) material in the latter part of Earth's history, to explain the observed τPb. However, simulations in which fractionation is controlled by a single set of partition coefficients throughout the model runs fail to reproduce τPb and the observed scatter in Pb isotope ratios. We build on these models with 3D mantle convection simulations including parameterizations for melting, U recycling from the continents and preferential removal of Pb from subducted oceanic crust. We find that both U recycling after the great oxygenation event and Pb extraction after the onset of plate tectonics, are required in order to fit the observed gradient and scatter of both the ²⁰⁷Pb/²⁰⁴Pb—²⁰⁶Pb/²⁰⁴Pb and ²⁰⁸Pb/²⁰⁴Pb—²⁰⁶Pb/²⁰⁴Pb arrays. Unlike much previous work, our model does not require accumulations of subducted oceanic crust to persist at the core‐mantle boundary for long periods of time in order to match geochemical observations.
... The mantle dynamics code [22][23] [24][25] is a 3D, finite-element model that uses a mesh comprised of hexagonal prism elements constructed from the regular icosahedron. The mantle's silicate material is treated as a viscous fluid. ...
... If the Clapeyron slope is positive and negative, respectively, the phase transition enhances and impedes mantle convection [1]. Many geodynamicists simulated mantle convections by considering the Clapeyron slopes to demonstrate their impact on mantle dynamics and evolution [2][3][4][5][6][7][8][9][10]. ...
The Clapeyron slope is the slope of a phase boundary in P–T space and is essential for understanding mantle dynamics and evolution. The phase boundary is delineating instead of balancing a phase transition’s normal and reverse reactions. Many previous high pressure–temperature experiments determining the phase boundaries of major mantle minerals experienced severe problems due to instantaneous pressure increase by thermal pressure, pressure drop during heating, and sluggish transition kinetics. These complex pressure changes underestimate the transition pressure, while the sluggish kinetics require excess pressures to initiate or proceed with the transition, misinterpreting the phase stability and preventing tight bracketing of the phase boundary. Our recent study developed a novel approach to strictly determine phase stability based on the phase equilibrium definition. Here, we explain the details of this technique, using the post-spinel transition in Mg2SiO4 determined by our recent work as an example. An essential technique is to observe the change in X-ray diffraction intensity between ringwoodite and bridgmanite + periclase during the spontaneous pressure drop at a constant temperature and press load with the coexistence of both phases. This observation removes the complicated pressure change upon heating and kinetic problem, providing an accurate and precise phase boundary.
... The second model (Müller et al., 2018) constrains a three-fold approach of convection-driven dynamic topography developed for the continental interior, its passive margins, and their hinterlands through time. The third model (Rubey et al., 2017) based on a quantitative framework to make a linkage between the mantle convection (using code ''TERRA" of Baumgardner, 1985;Bunge et al., 1997;Davies and Davies, 2009;Wolstencroft et al., 2009;Davies et al., 2013;Wolstencroft and Davies, 2017) with the plate tectonics and basin evolution. The fourth model (Flament et al., 2013) is likely the residual topography calculated by removing the isostatic effect of sediments, water, ice, and lithosphere/crust from the observed topography. ...
The sediments and their temporal and spatial distribution on Earth are the archives of interaction between deep and surface processes. Numerous simulations have attempted to decipher the evolution of landscape and sedimentary basins. However, it remains a challenge to couple paleogeographic evolution with deep geodynamics, especially at the continental margins such as the Northern South China Sea Margin (NSCSM). Here we employ a numerical simulating tool (Badlands) for the tectono-sedimentary evolution of the NSCSM to test and compare our restored interpolated paleo-bathymetries with previously published paleogeography. Based on the subsequent reconstruction of cumulative sediment thicknesses at the distal boundary of the continental margin, we found that the erodibility and elastic thickness of the lithosphere are the best-fitted and the most promising parameters to isostatically deal with deep dynamics in controlling regional topography. According to our findings, the Early Oligocene and Early Miocene are two stages of high erodibility and fast uplifting in the source provenances or weathering regions as well as lithospheric strengthening. In contrast, the lower values of elastic thickness indicate the Eocene rifting and Late Miocene reactivation of extension due to its lithospheric weakening, relative to the post-rift thickening.
Plain language summary: Transient deep and surface processes actively participated in the geography of Earth’s landscape and sediment distribution. It takes millions of years for these processes and remains challenging to investigate the past landscape evolution. However, in the era of numerical modelling and more advanced research, it is now possible to solve many scientific questions. This study also attempted to find paleogeographic core findings based on the combined effect of deep and surface processes using a numerical simulation tool, Badland. Finally, we establish an erosion as a surface leading factor and an elastic thickness of lithosphere and mantle convection as the deep dynamic factors, to effect the Earth’s surface landscape and link with the accumulation of sediments.
... Thermal effects are indirectly considered by assigning the proper physical properties (e.g., density and viscosity) to the different bodies (e. g., Mason et al., 2010;Capitanio, 2014;Capitanio et al., 2015;Király et al., 2016Király et al., , 2017Peral et al., 2020). The isothermal assumption is valid as long as the subduction velocities are greater than 10 mm/yr (Bunge et al., 1997;Wortel, 1982). The rheological behavior of the sublithospheric mantle (deep blue color), the Iberian and African continental blocks (reddish colors), and the Alboran-Tethys and Algerian-Tethys plate segments (light blue colors) are modelled as follows (Fig. 4b, c and d). ...
The geodynamic evolution of the Western Mediterranean related to the closure of the Ligurian-Tethys ocean is not yet fully resolved. We present a new 3D numerical model of double subduction with opposite polarities fostered by the inherited segmentation of the Ligurian-Tethys margins and rifting system between Iberia and NW Africa. The model is constrained by plate kinematic reconstructions and assumes that both Alboran-Tethys and Algerian-Tethys plate segments are separated by a NW-SE transform zone enabling that subduction polarity changes from SE-dipping in the Alboran-Tethys segment to NW-dipping in the Algerian-Tethys segment. The model starts about late Eocene times at 36.5 Ma and the temporal evolution of the simulation is tied to the geological evolution by comparing the rates of convergence and trench retreat, and the onset and end of opening in the Alboran Basin. Curvature of the Alboran-Tethys slab is imposed by the pinning of its western edge when reaching the end of the transform zone in the adjacent west-Africa continental block. The progressive curvature of the trench explains the observed regional stress reorientation changing from N-S to NW-SE and to E-W in the central and western regions of the Alboran Basin. The increase of the retreat rates from the Alboran-Tethys to the Algerian-Tethys slabs is compatible with the west-to-east transition from continental-to-magmatic-to-oceanic crustal nature and with the massive and partially synchronous calc-alkaline and alkaline magmatism.
... Leitch et al. (1991) studied convection with internal heating with pressure dependent thermal expansivity, whereas Travis and Olson (1994) showed how internal heating sources cause thermal turbulence. Bunge et al. (1997) did a comparative study on how pressure dependent viscosity, internal heating and phase change influence mantle convection at very high Rayleigh number. McNamara and Zhong (2005) investigated the effect of internal heating on mantle convection with temperature-dependent viscosity. ...
A mathematical model is considered for Rayleigh–Bénard convection of mantle where the viscosity depends strongly on both temperature and pressure defined in an Arrhenius form. The model is solved numerically for extremely large viscosity variations across a unit aspect ratio cell using a modified cut-off viscosity law, and steady solutions are obtained. The aim is to investigate the convection pattern with internal heating at a very high viscosity variation in the presence of high Rayleigh number. The study also investigates the relation between temperature dependent parameter and pressure dependent parameter in a basally heated convection cell. The numerical simulation is performed using the finite element method based PDE solver and the results are presented through figures, tables and graphs.
... Global 3-D spherical mantle convection models, and studies focussing on their application, are now in common use (e.g. Baumgardner, 1985;Tackley et al., 1993;Bunge et al., 1996Bunge et al., , 1997Zhong et al., 2000;Oldham and Davies, 2004;McNamara and Zhong, 2005;Choblet et al., 2007;Zhong et al., 2008;Tackley, 2008;Davies and Davies, 2009;Wolstencroft et al., 2009;Stadler et al., 2010;Tan et al., 2011;Kronbichler et al., 2012;Davies et al., 2013;Burstedde et al., 2013;Heister et al., 2017;Dannberg and Gassmoller, 2018;Stotz et al., 2018). However, the use of this geometry for calculations at a realistic convective vigour remains expensive. ...
Computational models of mantle convection must accurately represent curved boundaries and the associated boundary conditions of a 3-D spherical shell, bounded by Earth's surface and the core–mantle boundary. This is also true for comparable models in a simplified 2-D cylindrical geometry. It is of fundamental importance that the codes underlying these models are carefully verified prior to their application in a geodynamical context, for which comparisons against analytical solutions are an indispensable tool. However, analytical solutions for the Stokes equations in these geometries, based upon simple source terms that adhere to physically realistic boundary conditions, are often complex and difficult to derive. In this paper, we present the analytical solutions for a smooth polynomial source and a delta-function forcing, in combination with free-slip and zero-slip boundary conditions, for both 2-D cylindrical- and 3-D spherical-shell domains. We study the convergence of the Taylor–Hood (P2–P1) discretisation with respect to these solutions, within the finite element computational modelling framework Fluidity, and discuss an issue of suboptimal convergence in the presence of discontinuities. To facilitate the verification of numerical codes across the wider community, we provide a Python package, Assess, that evaluates the analytical solutions at arbitrary points of the domain.
... Richards and Lenardic (2018) detailed the importance of the Cathles parameter in determining the impact of a low viscosity upper-mantle layer on postglacial rebound, dynamic topography and the geoid. They also connected the Cathles parameter to previous studies that showed how depth variable mantle viscosity could affect convective wavelengths in the mantle (Bunge et al. 1996(Bunge et al. , 1997Tackley, 1996Tackley, , 2000Zhong and Zuber 2001;Lenardic et al. 2006;Busse et al. 2006) as well as the balance between plate-driving and resisting forces (Höink and Lenardic 2010;Höink et al. 2011Höink et al. , 2012. Those studies treated mantle viscosity structure as an intrinsic mantle property, that is, they considered upper-mantle viscosity to be a static value. ...
Previous studies have shown that a low viscosity upper mantle can impact the wavelength of mantle flow and the balance of plate driving to resisting forces. Those studies assumed that mantle viscosity is independent of mantle flow. We explore the potential that mantle flow is not only influenced by viscosity but can also feedback and alter mantle viscosity structure owing to a non-Newtonian upper-mantle rheology. Our results indicate that the average viscosity of the upper mantle, and viscosity variations within it, are affected by the depth to which a non-Newtonian rheology holds. Changes in the wavelength of mantle flow, that occur when upper-mantle viscosity drops below a critical value, alter flow velocities which, in turn, alter mantle viscosity. Those changes also affect flow profiles in the mantle and the degree to which mantle flow drives the motion of a plate analogue above it. Enhanced upper-mantle flow, due to an increasing degree of non-Newtonian behaviour, decreases the ratio of upper- to lower-mantle viscosity. Whole layer mantle convection is maintained but upper- and lower-mantle flow take on different dynamic forms: fast and concentrated upper-mantle flow; slow and diffuse lower-mantle flow. Collectively, mantle viscosity, mantle flow wavelengths, upper- to lower-mantle velocities and the degree to which the mantle can drive plate motions become connected to one another through coupled feedback loops. Under this view of mantle dynamics, depth-variable mantle viscosity is an emergent flow feature that both affects and is affected by the configuration of mantle and plate flow.
... All mantle convection calculations have been performed using the three-dimensional finite element code TERRA (Baumgardner 1985; Bunge et al. 1997). The computational domain has been discretized with a regular mesh using more than 80 million finite elements, for a horizontal resolution of about 30 km at the surface, decreasing to half that value at the CMB, while the radial resolution is 25 km throughout the mantle. ...
The adjoint method is a powerful technique to compute sensitivities (Fréchet derivatives) with respect to model parameters, allowing one to solve inverse problems where analytical solutions are not available or the cost to determine many times the associated forward problem is prohibitive. In Geodynamics it has been applied to the restoration problem of mantle convection—that is, to reconstruct past mantle flow states with dynamic models by finding optimal flow histories relative to the current model state—so that poorly known mantle flow parameters can be tested against observations gleaned from the geological record. By enabling us to construct time dependent earth models the adjoint method has the potential to link observations from seismology, geology, mineral physics and palaeomagnetism in a dynamically consistent way, greatly enhancing our understanding of the solid Earth system. Synthetic experiments demonstrate for the ideal case of no model error and no data error that the adjoint method restores mantle flow over timescales on the order of a transit time (≈100 Myr). But in reality unavoidable limitations enter the inverse problem in the form of poorly known model parameters and uncertain state estimations, which may result in systematic errors of the reconstructed flow history. Here we use high-resolution, 3-D spherical mantle circulation models to perform a systematic study of synthetic adjoint inversions, where we insert on purpose a mismatch between the model used to generate synthetic data and the model used for carrying out the inversion. By considering a mismatch in rheology, final state and history of surface velocities we find that mismatched model parameters do not inhibit misfit reduction: the adjoint method still produces a flow history that fits the estimated final state. However, the recovered initial state can be a poor approximation of the true initial state, where reconstructed and true flow histories diverge exponentially back in time and where for the more divergent cases the reconstructed initial state includes physically implausible structures, especially in and near the thermal boundary layers. Consequently, a complete reduction of the cost function may not be desirable when the goal is a best fit to the initial condition. When the estimated final state is a noisy low-pass version of the true final state choosing an appropriate misfit function can reduce the generation of artefacts in the initial state. While none of the model mismatches considered in this study, taken singularly, results in a complete failure of the recovered flow history, additional work is needed to assess their combined effects.
... As long as the model is isochemical, ALA/TALA implementations can include phase transitions in the background reference density profile, and they can even model the effects of a non-zero Clapeyron slope and the latent heat release or consumption by adding the corresponding terms in the temperature equation and in the buoyancy term (e.g. Christensen & Yuen 1985;Tackley et al. 1993;Bunge et al. 1997;Nakagawa & Tackley 2004;Nakagawa et al. 2009;. However, in chemically heterogeneous models, the depth range and density change of a phase transition depends on the chemical composition, which means that it cannot be included in a single reference profile. ...
Mantle convection and long-term lithosphere dynamics in the Earth and other planets can be treated as the slow deformation of a highly viscous fluid, and as such can be described using the compressible Navier–Stokes equations. Since on Earth-sized planets the influence of compressibility is not a dominant effect, density deviations from a reference profile are at most on the order of a few percent and using the full governing equations poses numerical challenges, most modelling studies have simplified the governing equations. Common approximations assume a temporally constant, but depth-dependent reference profile for the density (the anelastic liquid approximation), or drop compressibility altogether and use a constant reference density (the Boussinesq approximation). In most previous studies of mantle convection and crustal dynamics, one can assume that the error introduced by these approximations was small compared to the errors that resulted from poorly constrained material behaviour and limited numerical accuracy. However, as model parametrizations have become more realistic, and model resolution has improved, this may no longer be the case and the error due to using simplified conservation equations might no longer be negligible: while such approximations may be reasonable for models of mantle plumes or slabs traversing the whole mantle, they may be unsatisfactory for layered materials experiencing phase transitions or materials undergoing significant heating or cooling. For example, at boundary layers or close to dynamically changing density gradients, the error arising from the use of the aforementioned compressibility approximations can be the dominant error source, and common approximations may fail to capture the physical behaviour of interest. In this paper, we discuss new formulations of the continuity equation that include dynamic density variations due to temperature, pressure and composition without using a reference profile for the density. We quantify the improvement in accuracy relative to existing formulations in a number of benchmark models and evaluate for which practical applications these effects are important. Finally, we consider numerical aspects of the new formulations. We implement and test these formulations in the freely available community software aspect, and use this code for our numerical experiments.
... For N16-EB16, mantle circulation was modelled using the parallel finite element code TERRA (Bunge & Baumgardner 1995;Bunge et al. , 1997, which solves the conservation equations for mass, energy and momentum at infinite Prandtl number and very small Reynolds number (no inertial forces) in a spherical shell. A numerical mesh of more than 80 million finite elements was used to achieve a high resolution, which is a prerequisite for modelling global mantle flow at earth-like conditions. ...
Tomographic-geodynamic model comparisons are a key component in studies of the present-day state and evolution of Earth’s mantle. To account for the limited seismic resolution, ‘tomographic filtering’ of the geodynamically predicted mantle structures is a standard processing step in this context. The filtered model provides valuable information on how heterogeneities are smeared and modified in amplitude given the available seismic data and underlying inversion strategy. An important aspect that has so far not been taken into account are the effects of data uncertainties. We present a new method for ‘tomographic filtering’ in which it is possible to include the effects of random and systematic errors in the seismic measurements and to analyse the associated uncertainties in the tomographic model space. The ‘imaged’ model is constructed by computing the generalized-inverse projection (GIP) of synthetic data calculated in an earth model of choice. An advantage of this approach is that a reparametrization onto the tomographic grid can be avoided, depending on how the synthetic data are calculated. To demonstrate the viability of the method, we compute traveltimes in an existing mantle circulation model (MCM), add specific realizations of random seismic ‘noise’ to the synthetic data and apply the generalized inverse operator of a recent Backus–Gilbert-type global S-wave tomography. GIP models based on different noise realizations show a significant variability of the shape and amplitude of seismic anomalies. This highlights the importance of interpreting tomographic images in a prudent and cautious manner. Systematic errors, such as event mislocation or imperfect crustal corrections, can be investigated by introducing an additional term to the noise component so that the resulting noise distributions are biased. In contrast to Gaussian zero-mean noise, this leads to a bias in model space; that is, the mean of all GIP realizations also is non-zero. Knowledge of the statistical properties of model uncertainties together with tomographic resolution is crucial for obtaining meaningful estimates of Earth’s present-day thermodynamic state. A practicable treatment of error propagation and uncertainty quantification will therefore be increasingly important, especially in view of geodynamic inversions that aim at ‘retrodicting’ past mantle evolution based on tomographic images.
... Jull and Kelemen [96] reported that a density contrast between the lowermost lithosphere and the underlying asthenosphere as low as 50 kg m -3 is sufficient to trigger "convective instability" that is, the denser material sinks under the action of gravity, while the lighter one, behaving like a fluid, rises to take its place. Gradients of the gravity acceleration due to density contrast, however, may be not sufficient to give rise to departures from metastable hydrostatic equilibrium, as they can be damped, for instance, by the viscosity of the lighter medium [97]. Moreover, viscosity, as known, is closely related to composition, temperature and stress conditions of the system (e.g., strain rate proportional to deviatoric stress, raised to a power n [98]), as can be seen from the classic works by Rayleigh [99] and Taylor [100] (e.g., "Rayleigh-Taylor instability" [98]). ...
Geochemical characteristics of middle ocean ridge basalts (MORBs) testify partial melting of spinel-peridotite mixed with a few amounts of garnet-pyroxenite. The latter can be considered either autochthonous products of the crystallization of partial melts in the sub-oceanic mantle or allocthonous recycled crustal materials originated in subduction contexts. Here we suggest the “autocthnous recycled” origin for garnet-pyroxenites. Such a hypothesis derives from the study of garnet-bearing pyroxenite xenoliths from the Hyblean Plateau (Sicily). These consist of Al-diopside, pyralspite-series garnet, Al-spinel and Al-rich orthopyroxene. Trace element distribution resembles an enriched MORB but lower chromium. Major-element abundances closely fit in a tschermakitic-horneblende composition. Assuming that a high-Al amphibolite was formed by hydrothermal metasomatism of a troctolitic gabbro in a slow-spreading ridge segment, a transient temperature increasing induced dehydroxilization reaction in amphiboles, giving Al-spinel-pyroxenite and vapor as products. Garnet partially replaced spinel during an isobaric cooling stage. Density measurements at room conditions on representative samples gave values in the range 3290–3380 kg m–3. In general, a density contrast ≥300 kg m–3 can give rise to convective instability, provided a sufficient large size of the heavy masses and adequate rheological conditions of the system. Garnet-pyroxenite lumps can therefore sink in the underlying mantle, imparting the “garnet geochemical signature” to newly forming basaltic magma.
... Global 3-D spherical mantle convection models, and studies focussing on their application, are becoming more common (e.g. Baumgardner, 1985;Tackley et al., 1993;Bunge et al., 1996Bunge et al., , 1997Zhong et al., 2000;Oldham and Davies, 2004;McNamara et al., 1993;Kramer et al., 2012), with benchmarking typically undertaken against solutions from other comparable codes, for incompressible (Blankenbach et al., 1989;Travis et al., 1990;Busse et al., 1994;van Keken et al., 1997van Keken et al., , 2008Tosi et al., 2015) and compressible (King et al., 2009) convection. The number of comparable studies, within a 2-D cylindrical or 3-D spherical geometry, however, is more limited. ...
Computational models of mantle convection must accurately represent curved boundaries and the associated boundary conditions within a 3-D spherical shell, bounded by Earth's surface and the core-mantle boundary. This is also true for comparable models in a simplified 2-D cylindrical geometry. It is of fundamental importance that the codes underlying these models are carefully verified prior to their application in a geodynamical context, for which comparisons against analytical solutions are an indispensable tool. However, analytical solutions for the Stokes equations in these geometries, based upon simple source terms that adhere to natural boundary conditions, are often complex and difficult to derive. In this paper, we present the analytical solutions for a smooth polynomial source and a delta-function forcing, in combination with free-slip and zero-slip boundary conditions, for both 2-D cylindrical and 3-D spherical shell domains. We study the convergence of the Taylor Hood (P2-P1) discretisation with respect to these solutions, within the finite element computational modelling framework Fluidity, and discuss an issue of suboptimal convergence in the presence of discontinuities. To facilitate the verification of numerical codes across the wider community, we provide a python package, Assess, that evaluates the analytical solutions at arbitrary points of the domain.
... The depth-dependent case differs from the iso-viscous case through a lowering of the upper mantle viscosity by a factor of 30. As per previous 3D spherical studies (Bunge et al., 1996(Bunge et al., , 1997, this leads to an increase in the wavelength of convection. It also increases the sub-adiabatic temperature gradient in the mantle. ...
... In the first the ancestral Hawaiian plume is drawn at mid-mantle depths toward rapid spreading at the Late Cretaceous Kula/Pacific ridge 6 , a process supported by numerical simulations 49 . As spreading wanes, and eventually ceases altogether, the ridge influence on the plume diminishes. ...
Controversy surrounds the fixity of both hotspots and large low shear velocity provinces (LLSVPs). Paleomagnetism, plate-circuit analyses, sediment facies, geodynamic modeling, and geochemistry suggest motion of the Hawaiian plume in Earth's mantle during formation of the Emperor seamounts. Herein, we report new paleomagnetic data from the Hawaiian chain (Midway Atoll) that indicate the Hawaiian plume arrived at its current latitude by 28 Ma. A dramatic decrease in distance between Hawaiian-Emperor and Louisville chain seamounts between 63 and 52 Ma confirms a high rate of southward Hawaiian hotspot drift (~47 mm yr-1), and excludes true polar wander as a relevant factor. These findings further indicate that the Hawaiian-Emperor chain bend morphology was caused by hotspot motion, not plate motion. Rapid plume motion was likely produced by ridge-plume interaction and deeper influence of the Pacific LLSVP. When compared to plate circuit predictions, the Midway data suggest ~13 mm yr-1 of African LLSVP motion since the Oligocene. LLSVP upwellings are not fixed, but also wander as they attract plumes and are shaped by deep mantle convection.
... The lowest viscosity value is present in the asthenosphere. This layering of mantle viscosity leads to long-wavelength flow, i.e. large convection cells with large horizontal extent (Bunge, Richards & Baumgardner 1996, 1997Tackley 1996;Zhong et al. 2000;Lenardic, Richards & Busse 2006;Höink & Lenardic 2008). Consequently, the plate will move longer distances with the convective flow until it encounters impeding flow structures associated with downwellings, resulting in a decreased chance of being impeded. ...
Continents exert a thermal blanket effect to the mantle underneath by locally accumulating heat and modifying the flow structures, which in turn affects continent motion. This dynamic feedback is studied numerically with a simplified model of an insulating plate over a thermally convecting fluid with infinite Prandtl number at Rayleigh numbers of the order of $10^{6}$ . Several plate–fluid coupling modes are revealed as the plate size varies. In particular, small plates show long durations of stagnancy over cold downwellings. Between long stagnancies, bursts of velocity are observed when the plate rides on a single convection cell. As plate size increases, the coupled system transitions to another type of short-lived stagnancy, in which case hot plumes develop underneath. For an even larger plate, a unidirectional moving mode emerges as the plate modifies impeding flow structures it encounters while maintaining a single convection cell underneath. These identified modes are reminiscent of some real cases of continent–mantle coupling. Results show that the capability of a plate to overcome impeding flow structures increases with plate size, Rayleigh number and intensity of internal heating. Compared to cases with a fixed plate, cases with a freely drifting plate are associated with higher Nusselt number and more convection cells within the flow domain.
... As the ridge upwelling diminished, the plume conduit may have returned to a more vertical geometry, resulting in the hotspot motion pattern detected on the surface. Numerical simulations and experimental analogs suggest this is possible (Bunge et al, 1997;Tarduno et al., 2009). The second explanation invokes a "bottom-up" control. ...
Hotspots tracks—chains of volcanic edifices arising from deep mantle upwellings—were once thought to solely record plate motion. Results of ocean drilling expeditions have led to a transformative change: it is now recognized that these tracks can also reflect the motion of hotspots in Earth’s mantle. When hotspots move, their paths can provide insight into the nature of the mantle and the history of convection.
... We verify the newly derived equations with twin experiments, where we retrodict the evolution of a reference model (reference twin) with thermochemical anomalies. The forward mantle convection equations are solved with the three-dimensional spherical mantle convection code TERRA [41,42] for compressible, isoviscous flow implemented on a cluster dedicated to largescale geophysical capacity computing [43]. Table 1 are the model parameters employed in this study. ...
The adjoint method is an efficient way to obtain gradient information in a mantle convection model relative to past flow structure, allowing one to retrodict mantle flow from observations of the present-day mantle state. While adjoint equations for isochemical mantle flow have been derived for both incompressible and compressible flows, here we extend the method to thermochemical mantle flow models, and present thermochemical adjoint equations in the elastic-liquid approximation. We verify the method with twin experiments, and retrodict the flow history of a thermochemical reference model (reference twin) assuming for the final state, either a consistent thermochemical interpretation, using the thermochemical adjoint equations, or an inconsistent purely thermal interpretation, using the isochemical adjoint equations. The consistent simulation correctly retrodicts the flow evolution of the reference twin. The inconsistent case, instead, restores a false flow history whereby internal buoyancy forces and convectively maintained topography are overestimated. Because the cost function is reduced in either case, our results suggest that the adjoint method can be used to link assumptions on the role of chemical mantle heterogeneity to geologic inferences of dynamic topography, thus providing additional means to test hypotheses on mantle composition and dynamics. © 2018 The Author(s) Published by the Royal Society. All rights reserved.
... This presented a dynamic problem in that long-wavelength cells are not the norm at the level of convective vigor inferred for the Earth's mantle, that is, for Rayleigh numbers of order 10 7 -10 9 (Busse, 1985;Turcotte & Schubert, 1982). A potential solution to the long-wavelength flow problem came from numerical simulations of mantle convection that showed that a high-viscosity lower mantle below a low-viscosity upper mantle could lead to long-wavelength flow (e.g., Bunge et al., 1996Bunge et al., , 1997Hansen et al., 1993;Tackley, 1996;Zhang & Yuen, 1995;Zhong et al., 2000;Zhong & Zuber, 2001). Low-viscosity upper mantle is a coarse model analog of the asthenosphere, and the numerical results are consistent with the idea that an asthenosphere facilitates long-wavelength mantle convection. ...
A weak asthenosphere, or low-viscosity zone (LVZ), underlying Earth’s lithosphere has played an important role in interpreting isostasy, postglacial rebound (PGR), and the seismic LVZ, as well as
proposed mechanisms for continental drift, plate tectonics, and postseismic relaxation. Consideration of the resolving power of PGR, postseismic relaxation, and geoid modeling studies suggests a sublithospheric LVZ perhaps ~100–200 km thick with a viscosity contrast of ~100–1,000. Ab initio numerical models of plate-like boundary layer motions in mantle convection also suggest a key role for the LVZ. Paradoxically, a thinner LVZ with a strong viscosity contrast is most effective in promoting plate-like surface motions. These numerical results are explained in terms of the reduction in horizontal shear dissipation due to an LVZ, and a simple scaling theory leads to somewhat nonintuitive model predictions. For example, an LVZ causes stress magnification at the base of the lithosphere, enhancing plate boundary formation. Also, flow within the LVZ may be driven by the plates (Couette flow), or pressure-driven from within the mantle (Poiseuille flow), depending upon the degree to which plates locally inhibit or drive underlying mantle convection. For studies of the long-wavelength geoid, PGR, and mantle convection, a simple dimensionless parameter controls the effect of the LVZ. This “Cathles parameter” is given by Ct = η*(D/λ)3, where η* is the viscosity contrast, D is the thickness of the LVZ, and λ is the flow wavelength, emphasizing the tightly coupled trade-off between LVZ thickness and viscosity contrast.
... Note that ASPECT (Kronbichler et al., 2012;Heister et al., 2017), a relatively new code aimed at superseding CitcomS, can generate and use this type of mesh (Thieulot, 2017) but is not limited to it. -The icosahedral mesh (HS20) is composed of 20 triangular blocks (Baumgardner and Frederickson, 1985;Baumgardner, 1985) subdivided into triangles, which is used in the TERRA code (Bunge et al., 1996(Bunge et al., , 1997(Bunge et al., , 1998Davies et al., 2013). ...
I present in this work the GHOST (Geoscientific Hollow Sphere Tessellation) software which allows for the fast generation of computational meshes in hollow sphere geometries counting up to 100 million cells. Each mesh is composed of concentric spherical shells which are built out of quadrilaterals or triangles. I focus here on three commonly used meshes used in geodynamics/geophysics and demonstrate the accuracy of shell surfaces and mesh volume measurements as a function of resolution. I further benchmark the built-in gravity and gravitational potential procedures in the simple case of a constant density geometry and finally show how the produced meshes can be used to visualise the S40RTS mantle tomography model. The code is open source and is available on the GitHub sharing platform.
... The system is isothermal. This is equivalent to considering that the slabs are in a quasi-adiabatic condition, which is a good approximation as long as the subduction velocities are greater than 1 cm/year (Bunge et al., 1997;Wortel, 1982). ...
Active and fossil subduction systems consisting of two adjacent plates with opposite retreating directions occur in several areas on Earth, as the Mediterranean or Western Pacific. The goal of this work is to better understand the first‐order plate dynamics of these systems using the results of experimental models. The laboratory model is composed of two separate plates made of silicon putty representing the lithosphere, on top of a tank filled with glucose syrup representing the mantle. The set of experiments is designed to test the influence of the width of plates and the initial separation between them on the resulting trench velocities, deformation of plates and mantle flow. Results show that the mantle flow induced by both plates is asymmetric relative to the axis of each plate causing a progressive merging of the toroidal cells that prevents a steady state phase of the subduction process and generates a net outward drag perpendicular to the plates. Trench velocities increase when trenches approach each other and decrease when they separate after their intersection. The trench curvature of both plates increases linearly with time during the entire evolution of the process regardless their width and initial separation. The interaction between the return flows associated with each retreating plate, particularly in the inter‐plate region, is stronger for near plate configurations and correlates with variations of rollback velocities. We propose that the inferred first order dynamics of the presented analogue models can provide relevant clues to understand natural complex subduction systems.
Earth's style of planetary heat transport is characterized by plate
tectonics which requires rock strength to be reduced plastically in
order to break an otherwise stagnant lithospheric lid, and for rocks
to have a memory of past deformation to account for strain
localization and the hysteresis implied by geological sutures.
Here, we explore ~10^7 Rayleigh number, visco-plastic, 3-D
global mantle convection with damage. We show that oceanic
lithosphere-only models generate strong toroidal-poloidal power
ratios and features such as a mix of long-wavelength tectonic
motions and smaller-scale, back-arc tectonics driven by
downwellings. Undulating divergent plate boundaries can evolve to form
overlapping spreading centers and microplates, promoted and perhaps
stabilized by the effects of damage with long memory. The inclusion
of continental rafts enhances heat flux variability and toroidal
flow, including net rotation of the lithosphere, to a level seen in
plate reconstructions for the Cenozoic. Both the super-continental
cycle and local rheological descriptions affect heat transport and
tectonic deformation across a range of scales, and we showcase both
general tectonic dynamics and regionally applied continental breakup
scenarios. Our work points toward avenues for renewed analysis of
the typical, mean behavior as well as the evolution of fluctuations
in geological and model plate boundary evolution scenarios.
The existence of a thin, weak asthenospheric layer beneath Earth's lithospheric plates is consistent with existing geological and geophysical constraints, including Pleistocene glacio-isostatic adjustment, modeling of gravity anomalies, studies of seismic anisotropy, and post-seismic rebound. Mantle convection models suggest that a pronounced weak zone beneath the upper thermal boundary layer (lithosphere) may be essential to the plate tectonic style of convection found on Earth. The asthenosphere is likely related to partial melting and the presence of water in the sub-lithospheric mantle, further implying that the long-term evolution of the Earth may be controlled by thermal regulation and volatile recycling that maintain a geotherm that approaches the wet mantle solidus at asthenospheric depths.
A hologram is an image in which each area contains almost all the information about the entire system. It is a metaphor commonly used for complex systems in which the whole is bigger than the sum of the parts because of self-organization. And also the whole is smaller than the sum of the parts, since the collective organization limits the behavior of dynamic features. The tectonic evolution of the Earth is an emergent behavior of the lithosphere-mantle system, a witness of a program defined at the scale of rocks. Mod-eling the physics behind tectonics at a global scale became a reachable goal entering the 21st century. Geodynamicists developed numerical models of solid-sate convection with yielding, and reproduced some fundamentals of planetary tectonics. In the past 15 years, several groups in the world have used these models to investigate how continents drift, seafloor spreads and plates evolve. These emergent characteristics tell that the whole is bigger than the sum of the parts. Slabs, plumes, ridges, plates are interdependent and constrain each other. The whole is smaller than the sum of the parts. In this context, searching for causality relationships between tectonic features seems vain. In this chapter, I consider this point of view to describe how con-vection models with yielding have changed and still change our views on how tectonics shape the Earth. I finally propose an outlook about this window that remains half-opened.
A hologram is an image in which each area contains almost all the information about the entire system. It is a metaphor commonly used for complex systems in which the whole is bigger than the sum of the parts because of self-organization. And also the whole is smaller than the sum of the parts, since the collective organization limits the behavior of dynamic features. The tectonic evolution of the Earth is an emergent behavior of the lithosphere-mantle system, a witness of a program defined at the scale of rocks. Modeling the physics behind tectonics at a global scale became a reachable goal entering the 21st Century. Geodynamicists developed numerical models of solid-sate convection with yielding, and reproduced some fundamentals of planetary tectonics. In the past 15 years, several groups in the world have used these models to investigate how continents drift, seafloor spreads and plates evolve. These emergent characteristics tell that the whole is bigger than the sum of the parts. Slabs, plumes, ridges, plates are interdependent and constrain each other. The whole is smaller than the sum of the parts. In this context, searching for causality relationships between tectonic features seems vain. In this chapter, I consider this point of view to describe how convection models with yielding have changed and still change our views on how tectonics shape the Earth. I finally propose an outlook about this window that remains half-opened.
Firedrake is an automated system for solving partial differential equations using the finite-element method. By applying sophisticated performance optimisations through automatic code-generation techniques, it provides a means of creating accurate, efficient, flexible, easily extensible, scalable, transparent and reproducible research software that is ideally suited to simulating a wide range of problems in geophysical fluid dynamics. Here, we demonstrate the applicability of Firedrake for geodynamical simulation, with a focus on mantle dynamics. The accuracy and efficiency of the approach are confirmed via comparisons against a suite of analytical and benchmark cases of systematically increasing complexity, whilst parallel scalability is demonstrated up to 12 288 compute cores, where the problem size and the number of processing cores are simultaneously increased. In addition, Firedrake's flexibility is highlighted via straightforward application to different physical (e.g. complex non-linear rheologies, compressibility) and geometrical (2-D and 3-D Cartesian and spherical domains) scenarios. Finally, a representative simulation of global mantle convection is examined, which incorporates 230 Myr of plate motion history as a kinematic surface boundary condition, confirming Firedrake's suitability for addressing research problems at the frontiers of global mantle dynamics research.
Floating rather stably above the convective mantle, continents exert a thermal blanket effect on the mantle by locally accumulating heat and altering the flow structure, which in turn affect continent motion. This thermal-mechanic feedback has been captured by a simplified model―a thermally insulating plate floating freely over a bottom-heated convecting fluid. Previous studies have revealed that, as plate size increases, plate motion transitions from long-term stagnancy over cold downwelling to short-term stagnancy over hot upwelling, termed stagnant modes (SMs) I and II, respectively, and eventually to a unidirectionally moving mode (UMM) with no stagnancy. During SM I/II, the adjacent hot/cold plumes are observed to move toward/away from the plate. The present study quantifies these plume motions through analysis on a partially insulated loop model established for SM I and SM II, respectively. The model predictions are well verified by results of numerical simulation. Moreover, the dominant frequency of plate speed is shown to increase linearly with plate size, with an abrupt jump as the coupling mode transitions from SM I to SM II. This suggests that the enhanced thermal blanket effect with plate size enables an increasingly fast feedback between the slowdown of a plate and the strengthening of hot upwelling which boosts plate motion. When plate motion is boosted before it stops, the UMM emerges. The predicted critical plate size for the UMM is well validated by simulation results. Geological implications of the results for continent motion, continent–mantle coupling and particularly the migration of plume and subduction sites are discussed.
I propose a new analysis method for determining the intraplate stress in geophysical models using a series of numerical simulations of mantle convection in 3D spherical-shell geometry. In the present study, the intraplate stress was evaluated from numerically obtained velocity and stress fields of mantle, and quantitatively classified into nine types by analyzing the principal deviatoric stress axes and the “stress ratio,” which is a continuous parameter accounting for the stress regimes. The sensitivity of model parameters and physical conditions associated with the basic characteristics of mantle convection, such as internal heating ratio, viscosity stratification, and temperature-dependent viscosity of the mantle as well as viscoplastic rheology that causes plate-like surface motion, on the intraplate stress regimes were studied. The results demonstrated that the radial viscosity structure of the mantle interior strongly affected intraplate stress regimes, and the combination of increased viscosity in the lower mantle and the low-viscosity asthenosphere enhanced the pure strike-slip faulting regime in the stable part of plate interiors. The temporally averaged toroidal-poloidal ratio (T/P ratio) at the top surface of mantle convection with surface plate-like motion and the mantle's viscosity stratification generally ranged ~20–40%, which is comparable to the observed T/P ratio of present-day and past Earth. Under such Earth-like surface conditions, normal faulting regime with strike-slip component or strike-slip regime with normal faulting component, as well as pure strike-slip faulting regime, were broadly found in the stable parts of the plate interiors. From the definition of the stress regime in the present study, strike-slip faults on the real Earth are likely to occur where the strike-slip faulting component is large in the present models. The analysis method proposed herein is effective for evaluating the intraplate stress in research target regions, for which observation data is insufficient to determine the intraplate stress.
Firedrake is an automated system for solving partial differential equations using the finite element method. By applying sophisticated performance optimisations through automatic code-generation techniques, it provides a means to create accurate, efficient, flexible, easily extensible, scalable, transparent and reproducible research software, that is ideally suited to simulating a wide-range of problems in geophysical fluid dynamics. Here, we demonstrate the applicability of Firedrake for geodynamical simulation, with a focus on mantle dynamics. The accuracy and efficiency of the approach is confirmed via comparisons against a suite of analytical and benchmark cases of systematically increasing complexity, whilst parallel scalability is demonstrated up to 12288 compute cores, where the problem size and the number of processing cores are simultaneously increased. In addition, Firedrake's flexibility is highlighted via straightforward application to different physical (e.g. complex nonlinear rheologies, compressibility) and geometrical (2-D and 3-D Cartesian and spherical domains) scenarios. Finally, a representative simulation of global mantle convection is examined, which incorporates 230 Myr of plate motion history as a kinematic surface boundary condition, confirming its suitability for addressing research problems at the frontiers of global mantle dynamics research.
One major challenge in studies on mantle convection is to provide a physically consistent link between underlying geophysical hypotheses and predicted surface observables. Here, we first review recent developments targeted toward linking hypothetical temperature fields to seismic recordings in a quantitative way. By combining mantle circulation models with mineralogical thermodynamics and global 3‐D seismic wavefield simulations, synthetic traveltime residuals can be computed that correctly capture the various nonlinearities in the relation to the underlying temperatures. We then highlight the importance of taking uncertainties in the input parameters into account when comparing to real data. Specifically, we investigate the effects of the poorly constrained parameters related to mineral anelasticity on the temperature‐velocity conversion and the predicted traveltime residuals. The anelastic correction increases the temperature sensitivity of seismic velocities, particularly at high temperatures. Assuming maximum values plausible for the frequency dependence of attenuation and the activation enthalpy of the dissipative process, the temperature derivative of shear‐wave velocity increases by more than 100% at the top, and up to 70% at the bottom of the lower mantle in regions of elevated temperatures. The standard deviation of the seismic traveltime residuals, however, is larger only by about 30% compared to the anharmonic case in consequence of the multiple nonlinearity of the physical situation at hand.
We propose a new pressure-based structure-preserving (SP) and quasi asymptotic preserving (AP) staggered semi-implicit finite volume scheme for the unified first order hyperbolic formulation of continuum mechanics [1], which goes back to the pioneering work of Godunov [2] and further work of Godunov & Romenski [3] and Peshkov & Romenski [4]. The unified model is based on the theory of symmetric-hyperbolic and thermodynamically compatible (SHTC) systems [2], [5] and includes the description of elastic and elasto-plastic solids in the nonlinear large-strain regime as well as viscous and inviscid heat-conducting fluids, which correspond to the stiff relaxation limit of the model. In the absence of relaxation source terms, the homogeneous PDE system is endowed with two stationary linear differential constraints (involutions), which require the curl of distortion field and the curl of the thermal impulse to be zero for all times. In the stiff relaxation limit, the unified model tends asymptotically to the compressible Navier-Stokes equations.
The new structure-preserving scheme presented in this paper can be proven to be exactly curl-free for the homogeneous part of the PDE system, i.e. in the absence of relaxation source terms. We furthermore prove that the scheme is quasi asymptotic preserving in the stiff relaxation limit, in the sense that the numerical scheme reduces to a consistent second order accurate discretization of the compressible Navier-Stokes equations when the relaxation times tend to zero. Last but not least, the proposed scheme is suitable for the simulation of all Mach number flows thanks to its conservative formulation and the implicit discretization of the pressure terms.
The theory of plate tectonics is widely accepted by scientists and provides a robust framework with which to describe and predict the behavior of Earth’s rigid outer shell – the lithosphere – in space and time. Expressions of plate tectonic interactions at the Earth’s surface also provide critical insight into the machinations of our planet’s inaccessible interior, and allow postulation about the geological characteristics of other rocky bodies in our solar system and beyond. Formalization of this paradigm occurred at a landmark Penrose conference in 1969, representing the culmination of centuries of study, and our understanding of the “what”, “where”, “why”, and “when” of plate tectonics on Earth has continued to improve since. In this Centennial review, we summarize the major discoveries that have been made in these fields and present a modern-day holistic model for the geodynamic evolution of the Earth that best accommodates key lines of evidence for its changes over time. Plate tectonics probably began at a global scale during the Mesoarchean (c. 2.9–3.0 Ga), with firm evidence for subduction in older geological terranes accounted for by isolated plate tectonic ‘microcells’ that initiated at the heads of mantle plumes. Such early subduction likely operated at shallow angles and was short-lived, owing to the buoyancy and low rigidity of hotter oceanic lithosphere. A transitional period during the Neoarchean and Paleoproterozoic/Mesoproterozoic was characterized by continued secular cooling of the Earth’s mantle, which reduced the buoyancy of oceanic lithosphere and increased its strength, allowing the angle of subduction at convergent plate margins to gradually steepen. The appearance of rocks during the Neoproterozoic (c. 0.8–0.9 Ga) diagnostic of subduction do not mark the onset of plate tectonics, but simply record the beginning of modern-style cold, deep, and steep subduction that is an end-member state of an earlier, hotter, mobile lid regime.
The solid Earth system is characterized by plate tectonics, a low viscosity zone beneath plates (the asthenosphere), and long wavelength flow in the convecting mantle. We use suites of numerical experiments to show: 1) How long wavelength flow and the operation of plate tectonics can generate and maintain an asthenosphere, and 2) How an asthenosphere can maintain long wavelength flow and plate tectonics. Plate subduction generates a sub-adiabatic temperature gradient in the mantle which, together with temperature-dependent viscosity, leads to a viscosity increase from the upper to the lower mantle. This allows mantle flow to channelize in a low viscosity region beneath plates (an asthenosphere forms dynamically). Flow channelization, in turn, stabilizes long wavelength convection. The degree of dynamic viscosity variations from the upper to the lower mantle increases with the wavelength of convection and drops toward zero if the system transitions from plate tectonics to a single plate planet. The plate margin strength needed to initiate that transition increases for long wavelength cells (long wavelength flow allows plate tectonics to exist over a wider range of plate margin strength). The coupled feedbacks allow for a linked causality between plates, the asthenosphere, and the wavelength of mantle flow, with none being more fundamental than the others and the existence of each depending on the others. Under this hypothesis, the asthenosphere is defined by an active process, plate tectonics, which maintains it and is maintained by it and plate tectonics is part of an emergent, self-sustaining flow system that bootstraps itself into existence.
Resolution and covariance of global seismic tomography models are most often unknown quantities. However, there are many potential applications of these matrices in the broad solid Earth research community as well as more focused scientific groups including the nuclear explosion monitoring research community. In this study, we construct both the resolution and covariance matrices for the recent LLNL-G3D-JPS global joint model of P- and S-wave velocity. The global model consists of >1 million free parameters, creating matrices with >1 trillion elements. Given the scale of the problem and computational limitations, we used a custom method to calculated impulse responses at every node in the earth model and produced sparse, yet representative, resolution and covariance matrices that can be practically used for several real applications. We apply the matrices to real problems as example use cases. Utilizing the covariance matrix, we computed traveltime uncertainties for thousands of P waves emanating from (or coming to) specified points around the globe and constructed maps of the traveltime error to illustrate the variability of path-specific traveltime uncertainty. Utilizing the resolution matrix as a tomographic filter, we converted geodynamically derived renditions of Earth structure to images that may be visible through the often-distorted lens of seismic tomography. © The Author(s) 2019. Published by Oxford University Press on behalf of The Royal Astronomical Society.
Thermally driven convection within the earth’s mantle determines one of the longest time scales of our planet. Plate tectonics, the piecewise continuous movement of the earth’s surface, is the prime manifestation of this slow deformational process but ultimately all large scale geological activity and dynamics of the planet involves the release of potential energy in the mantle. Massively parallel supercomputers are now allowing us to construct models of mantle convection with unprecedented complexity and realism. Here we present results from an approach to parallel computation that relies on explicit message‐passing and distributed computing. Connecting workstations together as a single parallel machine over a network and using the parallel virtual machine software, we are able to perform computations in the hundreds of Mflops range with a demonstrated total parallel overhead of less than ten percent. We have run high‐resolution thermal convection calculations for the earth’s mantle on the Los Alamos 16‐node cluster of IBM RS/6000 workstations employing a finite element mesh with more than 1.3 million grid points. These results indicate this approach to parallel computing offers a practical and efficient means of utilizing a broad spectrum of parallel hardware. © 1995 American Institute of Physics.
We report preliminary results from a sequence of very high resolution simulations of the mantle convection process based upon an axisymmetric anelastically compressible spherical model with both Olivine-Spinel and Spinel-post Spinel phase transitions. Our analyses strongly suggest that at Earth-like Rayleigh numbers near 107 the circulation may be quite strongly layered, although a very long timescale transient exists that is characterized by brief excursions into the whole mantle state. We suggest the use of a new diagnostic tool that may be employed to determine whether or not the layered state is characteristic of the present day earth.
Past studies of plate driving forces have concluded that the forces due to subducted slabs in the upper mantle and those due to the thickening of the oceanic lithosphere are the principal driving forces. We reexamine the balance of driving forces for the present-day and extend our analysis through the Cenozoic, using an analytical torque balance method which accounts for interactions between plates via viscous coupling to the induced mantle flow. We use an evolving mantle density heterogeneity field based on the last 200 Myr. of subduction to drive plate motions, an approach which has proven successful in predicting the present-day mantle heterogeneity field. We find that for plausible upper mantle viscosities the forces due to subducted slabs in the Cenozoic and Mesozoic account for in excess of 90% of plate driving forces and those due to lithospheric thickening for less than 10%.
The pressure dependence of the thermal expansion coefficient, α, previously reported as (∂lnaα/∂lnV)T = 5.5 ± 0.5 is refined, using systematics in the volume dependence of (∂T/∂P)s measured for a large number of materials at high pressures and high temperatures. Densities of perovskite (PV) and magnesiowustite (MW) are calculated for lower mantle conditions using our new α(P,T), a room temperature finite strain equation, and recent data on the Mg-Fe partitioning in the PV-MW system. A lower mantle made entirely of PV with a molar ratio of Mg:Fe of 88:12 would be about 0.11 g/cm3 or 2.5% denser than this mixture, but this density would just be within the uncertainty in PREM. A change in chemistry at 660 km depth to a PV mantle requires a thermal boundary which would improve the match in the densities between PV and PREM. Recent measurements on melting of Fe, FeO, and FeS, however, suggest temperatures at the core-mantle boundary below 3500 K, which tends to favour a geotherm without a large thermal boundary at 600 km depth. -from Authors
Uniform velocity/density scaling has been used to invert the seismically inferred 3-dimensional structure of the whole mantle for the radial viscosity structure which best fits the geoid. 60-72 percent variance reductions are obtained for three different S-wave tomographic models. The resulting viscosity structures are remarkably similar, showing a high viscosity lid, a low viscosity zone in the transition region, and a high viscosity lower mantle. A resolution analysis indicates that the viscosity structure in the upper mantle is well resolved by the data. However, the resolution in the lower mantle is poorer. The models are in general agreement with previous studies except that the inversions prefer a low viscosity layer at 400-670 km as opposed to 100-400 km.
Recent convection calculations have demonstrated that an endothermic phase transition can greatly decrease the vertical flow through the transition in a convecting system, in some cases leading to a layered flow. Using reasonable estimates of both the Rayleigh number and Clapeyron slope of the spinel to perovskite plus magnesiowuestite phase change, these results suggest that the 670-km phase change has a strong effect on mantle convection. This so-called 'dynamic layering' phenomenon is further investigated with a compressible finite element code using a two-dimensional, Cartesian geometry. We find a weak sensitivity of the pattern of flow to the form of the equations, considering Boussinesq, extended Boussinesq, and anelastic compressible forms of the governing equations, assuming that the thermodynamic properties (thermal expansivity, heat capacity, and latent heat) remain constant. The pattern of flow, however, depends strongly on the initial conditions, boundary conditions and equation of state. We compare the simple equation-of-state formulations used in previous work with a self-consistent equation of state based on Debye and Birch-Murnaghan finite strain theory under a Mie-Grueneisen formulation. A thermal expansion coefficient that decreases monotonically with depth and is unaffected by changes in phase or temperature greatly enhances dynamic layering. This trend is reversed when the temperature, pressure, and phase dependence on thermodynamic properties such as thermal expansivity, entropy, and heat capacity is introduced.
Numerical models of mantle convection that incorporate the major mantle phase changes of the transition zone reveal an inherently three-dimensional flow pattern, with cylindrical features and linear features that behave differently in their ability to penetrate the 670-km discontinuity. Flow penetration across the 670-km phase change is strongly wavelength-dependent, with easy penetration at long wavelengths but strong inhibition at short wavelengths. Thus, when comparing numerical models with long-wavelength seismic tomography, diagnostics based on the density field are much more sensitive to the effects of phase transitions than those based on the velocity field. The amplitude of the geoid is not significantly affected by the partial layering, because the contribution from the strong heterogeneity in the transition zone is almost perfectly balanced by the contribution from deflection of the 670-km discontinuity. -from Authors
The seismic velocity anomalies resolved by seismic tomography are associated with variations in density that lead to convective flow and to dynamically maintained topography at the Earth's surface, the core--mantle boundary (CMB), and any interior chemical boundaries that might exist. The dynamic topography resulting from a given density field is very sensitive to viscosity structure and to chemical stratification. The mass anomalies resulting from dynamic topography have a major effect on the geoid, which places strong constraints on mantle structure. Almost 90% of the observed geoid can be explained by density anomalies inferred from tomography and a model of subducted slabs, along with the resulting dynamic topography predicted for an Earth model with a low-viscosity asthenosphere (ca. 10^(20) Pa s) overlying a moderate viscosity (ca. 10^(22.5) Pa s) lower mantle. This viscosity stratification would lead to rapid mixing in the asthenosphere, with little mixing in the lower mantle. Chemically stratified models can also explain the geoid, but they predict hundreds of kilometres of dynamic topography at the 670 km discontinuity, a prediction currently unsupported by observation. A low-viscosity or chemically distinct D" layer tends to decouple CMB topography from convective circulation in the overlying mantle. Dynamic topography at the surface should result in long-term changes in eustatic sea level.
There are two obvious forms of convection in the Earth’s mantle: Plate-scale flow with upwellings at mid-ocean ridges and downwellings at subduction zones, and narrow upwelling plumes from the deep mantle which give rise to volcanic hotspots. Hotspots have relative motions which are at least an order of magnitude less than relative plate motions, and these two styles of convection appear to be decoupled on timescales of ~100–200 m.y. Also, hotspot basalts have trace element and isotopic signatures of mantle source regions which have been isolated from the upper mantle mid-ocean ridge source for a significant fraction of Earth history. These observations are compatible with whole mantle convection models which include a viscosity increase with depth. Lower mantle strain (mixing) rates and horizontal motions at, e.g., the core-mantle boundary are sufficiently reduced if lower mantle viscosity is about 1–2 orders of magnitude greater than that of the upper mantle. Such models also explain the depth distribution and focal mechanisms of deep earthquakes in subducted slabs as well as the long-wavelength geoid highs over subduction zones. Relatively isoviscous, chemically stratified convection models can probably satisfy the constraints from hotspot fixity and geochemistry. However, none of these observations has been shown to be compatible with models of whole-mantle convection which do not include a substantial increase in viscosity with depth.
The boundary-value problem is discretized on several grids (or finite-element spaces) of widely different mesh sizes. Interactions between these levels enable us (i) to solve the possibly nonlinear system of $n$ discrete equations in $O(n)$ operations ($40n$ additions and shifts for Poisson problems); (ii) to conveniently adapt the discretization (the local mesh size, local order of approximation, etc.) to the evolving solution in a nearly optimal way, obtaining "$\infty$-order" approximations and low $n$, even when singularities are present. General theoretical analysis of the numerical process. Numerical experiments with linear and nonlinear, elliptic and mixed-type (transonic flow) problems--confirm theoretical predictions. Similar techniques for initial-value problems are briefly discussed.
A wide range of geophysical and geochemical observations pertaining to convection in the earth's mantle and the dynamics of the tectonic plates is discussed. It is inferred that the dominant model of mantle convection is a plate-scale flow and that the plates are an integral part of this flow. Upwelling buoyant plumes, that cause volcanic hotspots, are inferred to comprise a secondary model of convection arising from a relatively weak thermal boundary layer at the base of the mantle. A significant viscosity increase is inferred by perhaps two to three orders of magnitude, through the depth of the mantle, with a large part of this increase occurring through the transition zone. With ridges sampling the top of the mantle and plumes sampling the bottom, these features offer explanations for the main geochemical characteristics of, and differences between, mid-ocean ridge basalts and oceanic island (hotspot) basalts. -from Authors
Snapshots of the temperature T(r,φ,t), horizontal flow velocity u(r,φ,t), and radial flow velocity w(r,φ,t) obtained from numerical convection experiments of time-dependent flows in annular cylindrical geometry are taken to be samples of stationary, rotationally invariant random fields. For such a field f(r,φ,t), the spatio-temporal two-point correlation function Cff(r,r′,Δ,t*), is constructed by averaging over rotational transformations of this ensemble. To assess the structural differences among mantle convection experiments we construct three spatial subfunctions of Cff(r,r′,Δ,t*). Two-point correlation functions of the temperature field (T-diagnostics) and flow velocity fields (V-diagnostics) have been used to quantify some important aspects of mantle convection experiments. -from Authors
This paper discusses the stochastic analysis of spatially complex, time-dependent flows in spherical and cylindrical geometries where the reference states, internal heating rates, and boundary conditions are temporally invariant and rotationally symmetric. Snapshots of the aspherical temperature anomalies δT (r,Ω,t) from a single convection run are taken to be samples of a stationary, rotationally invariant random field, and the spatial two-point correlation function CTT(r,r′,Δ) is constructed by averaging over rotational transformations of this ensemble. Three subfunctions are extracted: the rms variation, the radial correlation function, RT(r,r′) = CTT(r,r′,0)/ σT(r)σT(r′), and the angular correlation function All three are useful in assessing the structural differences among mantle convection simulations, but the diagnostic properties of RT and its robustness with respect to low-pass filtering recommend it as a tool for testing stratification hypotheses against whole-mantle tomographic models.
Numerical modeling of mantle convection in a spherical shell with an
endothermic phase change at 670 km depth reveals an inherently
three-dimensional flow pattern, containing cylindrical plumes and linear
sheets which behave differently in their ability to penetrate the phase
change. The dynamics are dominated by accumulation of downwelling cold
material above 670 km depth, resulting in frequent avalanches of
upper-mantle material into the lower mantle. This process generates
long-wavelength lateral heterogeneity, helping to resolve the
contradiction between seismic tomographic observations and expectations
from mantle convection simulations.
Studies are made on typical features of convection current in a thin
viscous layer on a deep layer of high viscosity. The streamlines in this
case are concentrated but not closed in the upper thin layer. The
horizontal velocity is one sign in the upper layer and the other sign in
the lower. The horizontal mass flux in the upper layer is balanced by
the counter-horizontal mass flux in the lower layer. The aspect ratio of
the critical perturbation, referred to the whole layer thickness, is not
so different from that in the homogeneous layer case. The aspect ratio,
referred to the thickness of the thin upper layer in which the current
is concentrated, is large, about 30 in a case studied in the present
paper. The result will give a theoretical justification for the
existence of very flat convection cells supposed to exist in the mantle.
The free surface of the fluid is low (high) where the temperature is
lower (higher) and the vertical velocity is downward (upward). This may
explain the formations of guyots in the Pacific. A quantitative check of
this idea and estimations of physical parameters involved in the problem
are made.
The structure and time dependence of 3-D thermal convection in a volumetrically heated, infinite Prandtl number fluid is examined for high values of the Rayleigh number. The methods employed allow the numerical experiments to proceed for long-enough times to derive good estimates of mean and fluctuating parts of the structure. An iterative multirigid method to solve for the buoyant, incompressible viscous flow at each time step of the energy equation is a novel aspect of the methodology. A simple explicit time step of the energy equation is utilized that vectorizes well on serial computers and which is ideally suited to massively parallel computers. Numerical experiments were carried out for Rayleigh numbers from 3 × 106 to 3 × 107 in a cartesian region with a prescribed temperature at the top boundary and an adiabatic bottom boundary. Over this complete range of Rayleigh number, the flow structure consists dominantly of cold, nearly axisymmetric plumes that migrate horizontally sweeping off the cold thermal-boundary layer that forms at the top of the convecting fluid. Plumes disappear by coalescing with other plumes; new plumes are created by thermal-boundary-layer instability. Sheet plumes form only occasionally and do not penetrate to significant depths in the fluid. Plumes have sizes comparable to the thickness of the thermal-boundary layer and an average spacing comparable to the fluid depth. No persistent large-scale motion in the fluid can be identified. Its absence may reflect the large subadiabatic stratification that develops beneath the thermal-boundary layer as cold plumes penetrate to the bottom boundary without thermally equilibrating with surrounding fluid. We consider the possible implications for convection in planetary mantles and for the existence of plate tectonics.
The high-pressure transformation in MgSiO3 and those in the
spinel phases of compositions from Mg2SiO4 to
(Mg0.5Fe0.5)2SiO4 in the
Mg2SiO4-Fe2SiO4 system were
investigated using a uniaxial split-sphere apparatus. The phase
boundaries between ilmenite-perovskite in MgSiO3 and between
Mg2SiO4 spinel and the assemblage of
MgSiO3 perovskite and MgO periclase were determined to be
P(GPa) = 26.8-0.0025T(°C) and P(GPa) = 27.6-0.0028T(°C),
respectively, in the temperature range 1000-1600°C. The pseudobinary
diagrams for the system
Mg2SiO4-Fe2SiO4 were
determined at temperatures of 1100°C and 1600°C. It was
demonstrated that the magnesian spinel (with Fe/Mg + Fe < 0.22 at
1100°C and <0.26 at 1600°C) dissociates into perovskite and
magnesiowüstite within an extremely narrow pressure interval
(<0.15 GPa at 1600°C). The dissociation pressure was found to be
almost independent of iron content and to coincide to that at 670 km
depth within experimental uncertainties. These experimental results
indicate that the sharpness of the 670-km discontinuity may indeed be
due to this dissociation in a peridotitic or pyrolitic mantle. The
current status of our understanding of deep mantle mineralogy and
chemistry is discussed based on recent high-pressure and
high-temperature experiments.
A numerical model of steady-state, two-dimensional free convection within an internally heated fluid of variable viscosity has been developed primarily for application to the earth's upper mantle. The dimensionless free-convection energy and stream function equations include advection, the adiabatic gradient, viscous dissipation, radiogenic heat sources, boundary heat fluxes, variable diffusivity, and variable viscosity. Both the energy equation and the fourth-order stream function equation are solved numerically by alternating direction implicit (ADI) algorithms on a special nonuniform grid first suggested by Samarskii.The Reynolds number is negligibly small, the Rayleigh number exceeds 106, and the Prandtl number exceeds 1023. The numerical convergence, accuracy and reliability of the method are established by various numerical tests.
This report investigates the effects of temperature dependent viscosity on 3D compressible mantle convection by means of numerical simulations in cartesian geometry using a finite volume multigrid code.
Using a simple criterion for the deflection of a constant-viscosity upwelling or downwelling by an endothermic phase transition, the scaling of the critical phase buoyancy parameter Pcrit with the important lengthscales is obtained. The derived trends match those previously observed in time-dependent numerical simulations, implying that geometry is the dominant factor in determining the propensity to layering. For a sinusoidal temperature anomaly, Pcrit is found to be proportional to wavelength, so that a stronger phase change is required to stop longer wavelengths, in accord with observations from three-dimensional numerical simulations. For more realistic Gaussian upwelling and downwelling features, the dependence of Pcrit on the width of feature, spacing of features, depth of phase transition and width of phase transition are determined for idealized internally heated and basally heated systems. Narrow upwellings and downwellings are deflected more easily than broad ones, providing a first-order explanation for the increased propensity to layering as Rayleigh number is increased. Internal heating is found to strongly favor deflection, particularly when the phase change is at shallow depth. For basally heated systems, the depth of the phase transition is found to be relatively unimportant in determining the value of Pcrit for which both upwellings and downwellings are deflected. In contrast, for internally heated systems, a shallower phase transition strongly favors layering. Only weak dependence of Pcrit on the spacing of upwellings and downwellings is found. A narrower phase transition enhances deflection.
We present a new spherical harmonic expansion of the global hotspot distribution, with the contribution of each hotspot weighted by its buoyancy flux. The resulting power spectrum, normalized to that expected for a random distribution, has statistically significant peaks only at degrees 1 and 2. To explain this, we study a simple model for the origin of mantle plumes in the D(double prime) layer at the base of the mantle: the Rayleigh-Taylor instability of two superposed fluid layers with different viscosities and densities in a spherical shell. The wavelength of the fastest growing instability greatly exceeds the thickness of D(double prime) and increases as the viscosity contrast gamma across this layer increases. These results are consistent with the hotspot spectrum if gamma approximately 10(exp 6).
Numerical models of mantle convection that incorporate the major mantle phase of the transition zone reveal an inherently three-dimensional flow pattern, with cylindrical features and linear features that behave differently in their ability to penetrate the 670-km discontinuity. The dynamics are dominated by accumulation of cold linear downwellings into rounded pools above the endothermic phase change at 670-km depth, resulting in frequent 'avalanches' of upper mantle material into the lower mantle. The effect of the exothermic phase transition at 400 km depth is to reduce the overall degree of layering by pushing material through the 670-km phase change, resulting in smaller and more frequent avalanches, and a wider range of morphologies. Large quantites of avalanched cold material accumulate above the coremantle boundary (CMB), resulting in a region of strongly depressed mean temperature at the base of the mantle. The 670-km phase change has a strong effect on the temperature field, with three distinct regions being visible. The effect of the velocity field is very different. Flow penetration across the 670-km phase change is strongly wavelength-dependent, with easy penetration at long wavelengths but strong inhibition at short wavelengths. Thus, when comparing numerical models with long-wavelength seismic tomography, diagnostics based on the density field, such as the radial correlation function, are much more sensitive to the effects of phase transitions than those based on the velocity field.
Thermally isolated 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 whose 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.
We describe nonlinear time-dependent numerical simulations of whole mantle convection for a Newtonian, infinite Prandtl number, anelastic fluid in a three-dimensional spherical shell for conditions that approximate the Earth's mantle. Each dependent variable is expanded in a series of 4,096 spherical harmonics to resolve its horizontal structure and in 61 Chebyshev polynomials to resolve its radial structure. A semiimplicit time-integration scheme is used with a spectral transform method. In grid space there are 61 unequally-spaced Chebyshev radial levels, 96 Legendre colatitudinal levels, and 192 Fourier longitudinal levels. For this preliminary study we consider four scenarios, all having the same radially-dependent reference state and no internal heating. They differ by their radially-dependent linear viscous and thermal diffusivities and by the specified temperatures on their isothermal, impermeable, stress-free boundaries. We have found that the structure of convection changes dramatically as the Rayleigh number increases from 105 to 106 to 107. The differences also depend on how the Rayleigh number is increased. That is, increasing the superadiabatic temperature drop, T, across the mantle produces a greater effect than decreasing the diffusivities. The simulation with a Rayleigh number of 107 is approximately 10,000 times critical, close to estimates of that for the Earth's mantle. However, although the velocity structure for this highest Rayleigh number scenario may be adequately resolved, its thermodynamic structure requires greater horizontal resolution. The velocity and thermodynamic structures of the scenarios at Rayleigh numbers of 105 and 106 appear to be adequately resolved. The 105 Rayleigh number solution has a small number of broad regions of warm upflow embedded in a network of narrow cold downflow regions; whereas, the higher Rayleigh number solutions (with large T) have a large number of small hot upflow plumes embedded in a broad weak background of downflow. In addition, as would be expected, these higher Rayleigh number solutions have thinner thermal boundary layers and larger convective velocities, temperatures perturbations, and heat fluxes. These differences emphasize the importance of developing even more realistic models at realistic Rayleigh numbers if one wishes to investigate by numerical simulation the type of convection that occurs in the Earth's mantle.
We present an almost uniform triangulation of the two-sphere, derived from the icosahedron, and describe a procedure for discretization of a partial differential equation using this triangular grid. The accuracy of our procedure is described by a strong theoretical estimate, and verified by large-scale numerical experiments. We also describe a data structure for this spherical discretization that allows fast computation on either a vector computer or an asynchronous parallel computer.
Accurate models of the distribution of elastic heterogeneity in the Earth's mantle are important in many areas of geophysics. The purpose of this paper is to characterize and compare quantitatively a set of recent three-dimensional models of the elastic structure of the Earth, to assess their similarities and differences, and to analyze their fit to one class of data in order to highlight fruitful directions for future research. The aspherical models considered are the following: M84C (Woodhouse and Dziewonski, 1984), LO2.56 (Dziewonski, 1984), MDLSH (Tanimoto, 1990a), SH.10c.17 (Masters et al., 1992), and S12_WM13 (Su et al., 1994). Through much of the discussion, M84C and LO2.56 are combined into a single whole mantle model, M84C + LO2.56. The fit of each model to previously tabulated even degree normal mode structure coefficients taken from Smith and Masters (1989a) and Ritzwoller et al. (1988) for multiplets along the normal mode fundamental and first, second, and fifth overtone branches is also presented. Rather than concentrating on detailed comparisons of specific features of the models, analyses of these models are general and statistical in nature. In particular, we focus on a comparison of the amplitude and the radial and geographical distribution of heterogeneity in each model and how variations in each affect the fit to the normal mode observations. In general, the results of the comparisons between the models are encouraging, especially with respect to the geographical distribution of heterogeneity and in the fit to the normal mode data sensitive to the upper mantle and lowermost lower mantle. There remain, however, significant discrepancies in amplitude and in the radial distribution of heterogeneity, especially near the top of the upper mantle and near the top of the lower mantle. The confident use of these models to constrain compositional and dynamical information about the mantle will await the resolution of these discrepancies. The factors that may be responsible for the differences in the models and/or for the misfit between the observed and predicted normal mode data are divided into two types: intrinsic (or procedural) and extrinsic (or structural). We discuss only three extrinsic factors at length here, including errors in the reference crustal models, unmodeled topography on discontinuities in the interior of the mantle, and errors in the assumed relationships between shear (υs) and compressional (υp) heterogeneity.
Lateral variations in seismic velocity (through its dependence on temperature) can easily be generated at the gravest harmonics, including degrees one and two, by the dynamic interaction between plates and convection. Models of thermal convection with a single non-subducting plate have been formulated in a cylindrical geometry. Plates of width one to four times times the thickness of the convecting region strongly modulate the flow by being pushed over cold downwellings and inhibiting cooling of the fluid beneath. During rapid motion off of hot regions, a large-scale pattern of shear is developed causing small uprising limbs to be swept into the largest upwellings. Both insulation and plume-plume collisions pump energy into the lower wavenumber harmonics.
We compute the value of the thermal expansivity α over a wide range of compression (η ≡ V / V 0 , 0.6 ≤ η ≤ 1.0) and temperature (300 K ≤ T ≤ 2500 K). Three methods are combined to find α. We utilize: 1) the high P,T data base from the PIB ab initio model for MgO, which specifies the Helmholtz energy and the volume and temperature derivatives; 2) the measured thermal expansivity and its various pressure and temperature derivatives; and 3) several thermodynamic identities relating T and V dependence of various physical properties.
We present a simple equation relating α to η along isochores and suggest values of α. for MgO over V, P, T conditions, including those of the earth's lower mantle. The parameters in the equation are evaluated by using the ab initio data base. We find that α varies from about 1.40α a to 0.40α a along a geotherm through the upper and lower mantle, where α a is α at ambient conditions.
The equations of motion are solved numerically for a Boussinesq fluid with infinite Prandtl number in a square 2-D box where the viscosity increases with depth. Three heating modes are employed: bottom heating, internal heating, and half bottom and half internal heating. In all cases the boundaries are free slip. The range of Rayleigh numbers employed is 104-107. The viscosity increases as 10β (1 - y), where y is distance measured from the bottom upwards and β is a free parameter. In the bottom heated cases, the convective velocities slow near the bottom and result in a large temperature drop between*** the bottom boundary and interior compared with the top boundary and the interior. This results in increased buoyancy in the ascending limb. In the internally heated case, the flow in the top half of the box resembles Rayleigh-Bènard convection and in the bottom half it approaches a conductive thermal regime for β greater than about 2. In this case the top surface heat flux decays from ascending to descending limb and the ascending and descending limbs become more equal in their buoyancy. Increasing β decreases the efficiency of heat transport, but has little effect on the exponents of Nu-Ra and Pe-Ra relations. There is a larger decrease in heat transport efficiency for a given β in the bottom heated case compared to the internally heated case.
The thermal expansitivities (α) of MgO and high-pressure phases of CaO, CaMgSi2O6, and Fe at ultra high pressure are obtained by comparing existing shock compression and temperature measurements of 300 K compression curves constructed from ultrasonic elasticity and static compression data. For MgO, an expression for α is presented. Using this expression, the thermal expansitivity of MgO is 28-32×10-6K-1 at the pressure of the top of the lower mantle and 10-16×10-6K-1 at its base (at 2000 K). New data for α of δ-Fe, together with an inner core temperature of 6750 K, constrain the density of the inner core to be 5±2% less than the density of ε-Fe, implying the inner core contains a light element. -from Authors
The Haskell [1935] value of 1021 Pas for mantle viscosity is a classic and enduring constraint on the rheology of the Earth's interior. We revisit this inference using spherically symmetric, self-gravitating, Maxwell viscoelastic Earth models. Our inference is based on both forward and inverse analyses of decay times associated with uplift at two sites considered by Haskell, Angerman River, Sweden, and Oslo, Norway, rather than the raw relative sea level (RSL) data at these sites. We demonstrate that predictions of the decay time associated with the Angerman River data are insensitive to variations in both the late Pleistocene ice load history and the lithospheric thickness of the Earth model (the predictions at Oslo are sensitive to both these inputs), and hence decay times at this site provide a remarkably robust constraint on mantle viscosity. We derive a constraint on the ``average'' viscosity of the mantle of 0.65-1.10×1021 Pas, where the ``average'' resolved by the data encompasses a region which extends from the base of the lithosphere to a depth of near 1400 km. This indicates that many previous analyses which have invoked the Haskell value of 1021 Pas as a constraint on the average upper mantle (i.e., above 670 km depth) viscosity alone have misinterpreted the resolving power of the inference. Furthermore, our analysis indicates that a number of apparently contradictory inferences of viscosity based on Fennoscandian data satisfy the new, rigorous, interpretation of the Haskell constraint. Finally, we demonstrate how the ambiguity in the upper mantle/lower mantle viscosity contrast associated with the Haskell ``average'' may be reduced by invoking decay time constraints estimated from RSL curves in Hudson Bay. A preliminary inversion of decay times at Angerman River and Richmond Gulf (in Hudson Bay) suggests a contrast of approximately an order of magnitude between the average viscosities of the upper mantle and the top 1000 km of the lower mantle; however, a conclusive analysis in this regard must await the determination of consistent decay time estimates for the Hudson Bay region.
Recent experiments on the melting temperatures of perovskite have
indicated a high melting temperature in the lower mantle. This suggests
that a creep law with an activation enthalpy, that increases strongly
with depth, should be employed for the lower mantle rheology. We have
examined the dynamical consequences of employing an Arrhenius type of
dependence in a Newtonian flow law under the Weertman assumption
relating activation enthalpy to the variation of the melting temperature
with pressure. We have employed finite element techniques to model both
steady state and time-dependent flows for such type of rheology in an
axisymmetric spherical- shell model. An outstanding dynamical feature
from these models is the presence of a local viscosity maximum with a
magnitude of around 1023Pa · s for Earthlike surface
Nusselt numbers between 10 and 15. This viscosity maximum is found in
the middle of the lower mantle and is a site of high deviatoric stresses
between 10 and 100 MPa. For the surface Rayleigh numbers examined
(104 to 105) the flows developed are not strongly
time-dependent and often tend to a steady state. In spite of the
presence of internal heating with chondritic strength, a few large,
relatively stationary plumes are found in the lower mantle, while the
rest of the lower mantle circulation is being driven by the more
vigorous upper mantle flow. There are no small-scale instabilities
developed in the D” layer of these models, thus suggesting that
any small-scale lateral heterogeneities existing there may have chemical
origins.
Numerical simulations have been performed using a multiphase, anelastic,
axisymmetric spherical, mantle convection model as part of an ongoing
effort to explore the ability of the endothermic phase transition at 660
km depth to cause the circulation to assume a layered style. In
particular, model solutions have been constructed for a Rayleigh number
of 10(exp 7), internal heating corresponding to 50% heating from within
and 50% heating from below, and Clapeyron slopes for the 410-km and
660-km phase boundaries set to +3.0 and -2.8 MPa/K, respectively. In
this regime the flow exhibits a substantial degree of radial layering
wherein the radial mass flux is reduced significantly at 660-km depth.
This layered regime of flow is episodically disrupted by massive
localized avalanches of fluid across the 660-km boundary that recur at
intervals separated by hundreds of millions of years. The degree of
layering is related to the magnitude of the 660-km phase boundary
deflection away from its average depth. In these Earth-like simulations
we find that the average magnitude of such phase boundary deflections is
similar to the average magnitude of seismically observed deflections of
this horizon.
The interaction between large-scale thermal structure of different wavelength and rafts and the effect of internal heating are investigated by using results from a series of numerical models in which internal heating, Rayleigh number, and raft size are varied. The results consistently show that raft motion is aperiodic and that significant large-scale lateral heterogeneity exists within and outside thermal boundary layers. The results also show that the continentlike raft is important for developing the long-wavelength structure.
A two-dimensional time-dependent numerical model of convection in a cylindrical annulus is described. Model results are presented and compared with similar models in planar geometry. In curvilinear geometry the upper (outer) boundary has a larger surface area than does the lower (inner) boundary. This results in an asymmetry between the upper and lower thermal boundary layers which is not found in planar geometry. A boundary layer argument is developed for curvilinear geometry which accounts for differences in model predictions between the curvilinear and planar models. It is found that in cylindrical geometry the effects of curvature on Nusselt number, mean temperature, relative thickness of the inner and outer thermal boundary layers, and temperature drops across the thermal boundary layers can all be parameterized in terms of the fraction ƒ of the inner to outer radii of the cylindrical bounding surfaces.
Dynamical calculations indicate that plate tectonics are likely to have
a major effect on the initiation of plumes from the basal boundary layer
in the mantle and their subsequent evolution. Plume start because the
boundary layer is hotter than the overlying mantle and hence unstable to
perturbations. The growth rate of perturbations is too weakly dependent
on wavelength to control the geographic distribution of plumes. Instead,
the distribution of plumes is strongly influenced by the shape of large
perturbations associated with slabs and surface plates. Plumes evolve
because downward return flow is needed to replace the material tapped
from the boundary layer. This flow downwarps the top of the boundary
layer several hundred kilometers from the plume and thus restricts
further flow of hot material toward an active plume. The geometry of
this downwarp is strongly influenced by the initial relief of the top of
the boundary layer and hence by plate processes. The actual history of
plates and slabs is thus needed in numerical models for comparison with
real data on plume fluxes.
We present a gravitationally consistent method for calculating the ``kernel'' functions which relate global geophysical observables, such as the geoid, surface plate motions and core-mantle boundary (CMB) topography, to internal density heterogeneities in a viscous, compressible mantle possessing an arbitrary radial variation of viscosity. We show that the influence of finite mantle compressibility is substantial in the case of the predicted nonhydrostatic geoid, with the largest effects occurring at spherical harmonic degree l=2. On the basis of the geoid data we find that our best two-layer viscosity model possesses a factor of 9 jump in viscosity at 1200 km depth. We argue, however, that the inferences of viscosity obtained from the geoid data are sensitive to the model of internal density heterogeneity which is employed and that this sensitivity probably explains the differences between or own viscosity inferences and those obtained by others. Our viscous flow models possess a spherically symmetric viscosity distribution and they cannot therefore ``explain'' the toroidal component of the flow of the tectonic plates. We therefore present a new scheme for predicting plate motions based on an explicit coupling of poloidal and toroidal flows which we derive on the basis of the assumption that the individual plates are perfectly rigid. We show how this new description of surface plate motions induced by buoyancy forces in the mantle may be used to constrain the absolute value of the long-term (i.e., steady state) mantle viscosity.
High-pressure and high-temperature phase relations in the system Mg2SiO4-Fe2SiO4 were thoroughly reinvestigated at pressures up to 21 GPa and at temperatures of 1200°C and 1600°C using a uniaxial split-sphere apparatus. Both normal and reverse experiments were performed on the univariant reactions. Both exsolving and dissolving experiments were performed to determine the divariant loops at 1600°C. The run products were examined using microfocused X ray diffractometry, scanning electron microscopy, and electron probe microanalysis. The phase diagrams thus constructed were compared with those previously proposed using an experimental or thermochemical method. Phase changes in mantle olivine with Mg number of 89 occur in the following order: (alpha)-(alpha+beta)-(beta)-(beta+gamma)-(gamma). The alpha+beta loop narrows with increasing temperature. The applicability of an isochemical peridotite mantle model is discussed in the light of the location and thickness of the 400-km discontinuity.
Numerical models of plate-scale convection confined to the upper mantle
predict large deviations from observed ocean bathymetry, gravity, and
geoid, while whole mantle models yield first-order agreement with these
observations. The upper mantle models fail because there is insufficient
radioactivity in the upper mantle to explain the surface heat flux, the
upper mantle must therefore be heated mainly from below, the resulting
hot boundary layer generates buoyant material, and when this buoyant
material rises to the base of the lithosphère, it generates large
positive anomalies in topography (2 km), the geoid (30 m), and gravity
(50 mGal). The concept of the magnitude of a hotspot swell is
introduced: it is the rate at which new topography is created, expressed
as a volume per unit time, and is a measure of the buoyancy and heat
fluxes in the underlying plume. By this measure the major Pacific hots
pots dominate the Earth's total plume flux, but plumes account for less
than 10% of the Earth's total heat flux. This is comparable to the
amount of heat likely to be coming from the core and supports the idea
that plumes originate at the core mantle boundary and not at the 670-km
transition. These results indicate that mantle convection is dominated
by a plate-scale flow which penetrates throughout the mantle, with a
secondary mode involving plumes rising from a weak thermal boundary
layer at the bottom of the mantle.
The process by which subducted lithosphere is mixed by mantle convection is investigated in numerical calculations. The results show that the observed isotopic heterogeneity of mantle sources and their ancient (1-2 b.y.)
apparent ages are consistent with convective mixing. Passive tracers, which are introduced below "trenches," are efficiently dispersed, but nonetheless, heterogeneities in tracer density with a large range of length scales are observed to persist for 40 or more transit times (one
transit time is the time to travel the fluid depth with the boundary velocity). In particular, there is a strong tendency to form high-density folds of the tracer strings, which persist much longer than simple shearing indicates. The folds persist because there is a strong tendency
for material that enters the flow at the margins of cells to be transferred to adjacent cells, where it is "unmixed." When the simulations are scaled to the whole mantle, the tight clumps (folds) of tracers are shown to persist for
up to 1-2 b .y. There is also a tendency for large-scale convection cells to remain isolated from recycled material for 1-2 b.y. These results are consistent with the significant chemical heterogeneity of the mantle as revealed by isotopic studies of oceanic basalts. Despite the spatial heterogeneity in tracer density, the average
time tracers remain in the box from subduction at trenches to sampling at ridges (i.e., the residence time) is well constrained and within 20% of the mean residence time expected from an analytic model in which tracers are assumed
to be sampled randomly. Model ages of the mantle that explicitly incorporate increased convection rates in the past and assume random sampling of heterogeneities bracket the - 2 b.y. apparent Pb-Pb and Rb-Sr isochrons of midocean ridge basalts and oceanic island basalts. The conclusion
of persistent spatial heterogeneity is different from the conclusions drawn from other studies. The different conclusions result, primarily, from our emphasis on the details of spatial variations as opposed to some average of the mixing, from a difference in flow unsteadiness, and from the different ways tracers have been introduced into the flow.
It is pointed out that the nature of the 670-km discontinuity in the
earth's mantle is of particular importance in geodynamics because of its
possible role of breaking mantle convection into two layers. In the
present investigation, the density change is considered as a change
which arises solely from a phase transition. The investigation is based
on a set of so-called 'extended Boussinesq equations' which include the
energy effects of adiabatic compression, latent heat, and frictional
heating. Attention is given to the constitutive equations and numerical
methods, the Boussinesq Limit Di=0, the finite dissipation number,
effective thermal expansivity, superplasticity, and layers of unequal
thickness.
The data from the International Seismological Centre bulletins for the years 1964-79 are used to derive a 3-D model of lateral variations of the P velocity in the lower mantle. Particular attention is given to the problem of weighting the individual observations in order to avoid, as much as possible, the bias due to the uneven distribution of sources and receivers. The resulting model shows a high level of perturbations just below the 670-km discontinuity and just above the core-mantle boundary. It also predicts well the large-scale pattern of observed travel time residuals for various source regions except for the distinct effects of the subduction zones. While the inferences made here with respect to the origin of the gravest orders of the geopotential field are not conclusive, they point the way in which results from seismology can be used to address some of the basic questions of geodynamics.-from Author
We have used a finite element model of time-dependent convection to determine the conditions for penetration of the subducted plate into the lower mantle. A temperature-dependent and non-Newtonian rheology is applied to achieve platelike behavior of the upper and sinking thermal boundary layer of convection. The 650-km discontinuity is taken as either a chemical or phase boundary or as a combination of both. It is represented by a marker chain which effects additional buoyancy when distorted out of its equilibrium position. When the compositional density contrast is greater than about 5%, the descending slab is deflected sidewards at the boundary and two-layer convection prevails. A resulting depression of the boundary in the range of 50-200 km should be detectable with seismic methods. Below 5% density difference the slab plunges several hundred kilometers into the lower mantle, and below 2% it will probably not stop before reaching the core-mantle boundary and extensive mixing would be expected. With a pure phase change a negative Clapeyron slope of about -6 MPa/K (-60 bar/K) is required to establish a type of "leaky" double-layer convection. A more moderate slope can aid a small compositional density difference to prevent slab penetration into the lower mantle. With the present uncertainties about the physical nature of the 650-km discontinuity, a variety of convective styles appears possible on dynamical grounds.
The concept of 'plate tectonics,' which has evolved from earlier theories of continental drift and sea floor spreading, has been remarkably successful in explaining the global distribution of seismicity [lsacks et al., 1968] and the linear magnetic anomalies symmetric with respect to the mid-ocean ridges [Vine and Matthews, 1963]. Although the relative motions of the major plates are fairly well known, the driving mechanism for plate motion is not clearly understood. A consideration of the energy necessary to maintain plate motion, as derived from the average worldwide seismic energy [Gutenberg, 1956], suggests that the most