Model Of Thermal Convection Research Articles

Thermal convection and the convective regime diagram in super-Earths

Numerical models of bottom-heated thermal convection of highly compressible fluid with strongly temperature-dependent viscosity are presented to understand how the Rayleigh number Ra and the temperature dependence of viscosity exert control over the regimes of thermal convection in massive super-Earths. Thermodynamic properties of mantle materials are pressure dependent, but other material properties including the viscosity are not. A stagnant lid develops along the surface of the planet, when the viscosity contrast across the mantle due to temperature dependence r exceeds 106 at high Rayleigh number relevant to super-Earths. The threshold in r, which increases with increasing Ra, is higher than that expected for the Earth from earlier Boussinesq models. The efficiency of convective heat transport measured by the Nusselt number Nu is considerably lower than that expected from Boussinesq models; Nu depends on Ra and r as Nu = 59 ⋅ r− 0.23 ⋅ (Ra/109)0.27, when r ≤ 105. Strong adiabatic compression significantly reduces the activity of hot ascending plumes especially at high r. At r relevant for super-Earths, hot ascending plumes lose their buoyancy on their way and hardly reach the surface boundary: hot spot volcanism due to ascending plumes is probably suppressed on super-Earths. The lithosphere is considerably thicker than that suggested by earlier Boussinesq models and is unlikely to show a plate-like behavior.

Thermal analysis and adiabatic calorimetry for early-age concrete members

This paper describes a procedure for thermal analysis on early-age concrete structures with experimentally obtained concrete adiabatic temperature rise and interface conductance between steel formwork and concrete. During hydration, concrete is considered in the analysis as non-homogeneous and the degree of hydration is dependent on both thermal loading and thermal properties development. A thermal convection model was implemented to simulate the external heat loss. Experimentally, two 1.2-m concrete cubes were constructed with embedded temperature sensors. The measured temperature time histories correlate well with the calculated values using a finite element model developed for this study. The modeling method and the experimental procedures developed are able to accurately predict the temperature evolution within an early-age concrete structure.

Exchange of stability in Cattaneo–LTNE porous convection

We investigate a problem of convection in local thermal non-equilibrium (LTNE) porous media. This is where the solid skeleton and the fluid may have different temperatures. However, due to increasing interest in thermal wave motion, especially at micro–nano-scales and particularly in solids, we analyse a model for thermal convection in a fluid saturated Darcy porous medium allowing the solid and fluid parts to be at different temperatures. This theory employs thermodynamics for the fluid based on Fourier’s law of heat conduction, whereas for the solid skeleton heat is transferred by means of the Cattaneo heat flux theory. Under appropriate conditions oscillatory convection is found whereas for the standard LTNE Darcy model this does not exist. In this article we concentrate on a transition which is found when the heat transfer interaction coefficient reaches a critical level and this leads to an exchange of stability for a certain porosity level.

The long‐wavelength geoid from three‐dimensional spherical models of thermal and thermochemical mantle convection

AbstractThe Earth's long‐wavelength geoid anomalies have long been used to constrain the dynamics and viscosity structure of the mantle in an isochemical, whole mantle convection model. However, there is strong evidence that the seismically observed large low shear velocity provinces (LLSVPs) in the lower mantle underneath the Pacific and Africa are chemically distinct and likely denser than the ambient mantle. In this study, we have formulated dynamically self‐consistent 3‐D spherical mantle convection models to investigate how chemically distinct and dense piles above the core‐mantle boundary may influence the geoid. Our dynamic models with realistic mantle viscosity structure produce dominantly spherical harmonic degree‐2 convection, similar to that of the present‐day Earth. The models produce two broad geoid and topography highs over two major thermochemical piles in the lower mantle, consistent with the positive geoid anomalies over the Pacific and African LLSVPs. Our geoid analysis showed that the bottom layer with dense chemical piles contributes negatively to the total geoid, while the layer immediately above the chemical piles contributes positively to the geoid, canceling out the effect of the piles. Thus, the bottom part of the mantle, as a compensation layer, has zero net contribution to the total geoid, and the thickness of the compensation layer is ~1000 km or 2 to 3 times as thick as the chemical piles. Our results help constrain and interpret the large‐scale thermochemical structure of the mantle using surface observations of the geoid and topography, as well as seismic models of the mantle.

Two-dimensional analytical model of dry air thermal convection

In the present work, the steady-state stationary dry air thermal convection in a lower atmosphere has been studied theoretically. The thermal convection was considered without accounting for the Coriolis force, and with only the vertical temperature gradient. The stream function has been analytically obtained within the framework of two-dimensional thermal convection model in the Boussinesq approximation with velocity divergence taken as zero. It has been shown that the stream function is symmetrical about the horizontal and vertical. The expressions for the horizontal and vertical air velocity components have been obtained. The maximal vertical velocities level is in the center of the convective cell where the horizontal air velocity component is equal to zero. It has been shown that the air parcel’s rotation period during the thermal convection is determined by the Brunt–Vaisala frequency. The expression for the maximal air velocity vertical component has been found. The dependence of the maximal air velocity vertical component on the overheat function at ground surface and on the atmosphere instability has been demonstrated. The expression for the pressure disturbance has been obtained. It has been demonstrated that at the points with maximal pressure disturbance the vertical velocity is equal to zero and the horizontal velocity is maximal. It has been found that the convection cell size depends on the atmosphere stability state.

Impact of phase change kinetics on the Mariana slab within the framework of 2-D mantle convection

Recent high-pressure and high-temperature experiments indicate that metastable olivine might persist in the cold core of a slab due to the low reaction rate of the olivine–wadsleyite phase transformation. Recent seismological observations detected a metastable olivine wedge that survives to a depth of 630km in the Mariana slab. To consider the problem of non-equilibrium phase transformation, we developed a two-dimensional (2-D) Cartesian numerical code that incorporates the effects of kinetics into a thermal convection model. We consider the kinetics of the 410-km olivine–wadsleyite and the 660-km ringwoodite–Pv+Mw phase transformations, including the effects of water content at the 410-km phase boundary. The latent heat release of the 410-km non-equilibrium phase transformations inside the slab is also considered. The results show positive correlations between some of the controlling parameters and the length of the metastable olivine wedge: the faster the subducting velocity, and the lower the water content, the deeper is the metastable olivine wedge. With increasing depth of phase transformation, the effect of latent heat release is enhanced: heating of, at most, 100°C occurs if olivine transforms into wadsleyite at a depth of approximately 570km in our model setting. Temperature increase due to the latent heat released stimulates further phase transformation, resulting in further temperature increase, acting as a positive feedback effect. We also attempt to explain the seismological observations by calculating the temperature and phase structures in the Mariana slab. If we assume that the age of the Mariana slab is 150Myr, the subduction velocity is 9.5cm/yr, phase transformation occurs from the grain boundary of the parental phase, and the water content is 250wt.ppm for a grain size of 1mm, 300wt.ppm for one of 5mm, and 100wt.ppm for intracrystalline transformation, then the metastable olivine wedge survives to a depth of 630km, which is in good agreement with the seismological observations. This suggests that the deeper portion of the Mariana slab is relatively dry. Assuming that the depression of the 660-km discontinuity by ∼20–30km within the Mariana slab, as is indicated by seismological observation, is explained by a combination of the depression caused by a negative Clapeyron slope in the cold slab and that due to the kinetics of the 660-km phase transformation, we obtain a gentle Clapeyron slope of −0.9MPa/K for the phase transformation from ringwoodite to Pv+Mw.

Structural stability for a thermal convection model with temperature-dependent solubility

We study a problem involving thermosolutal convection in a fluid when the solute concentration is subject to a chemical reaction in which the solubility of the dissolved component is a function of temperature. When the spatial domain is a bounded one in R2 we show that the solution depends continuously on the reaction rate using true a priori bounds for the solution when the chemical equilibrium function is an arbitrary function of temperature.

Effects of slab geometry and obliquity on the interplate thermal regime associated with the subduction of three-dimensionally curved oceanic plates

We investigated the relationships among slab geometry, obliquity, and the thermal regime associated with the subduction of oceanic plates using a three-dimensional (3D) parallelepiped thermal convection model. Various models with convex and concave slab shapes were constructed in the numerical simulation, and the temperature and mantle flow distributions were calculated. The results revealed that when the slab dip angle increases, or the obliquity of subduction becomes steeper, the interplate temperature decreases remarkably. Cooler (warmer) zones on the plate interface were identified from the modeling where there was a larger (smaller) subduction angle. Consequently, the interplate temperature distribution is partly controlled by the true subduction angle (TSA), which is a function of the slab dip angle and the obliquity of subduction. The rate of change of the interface temperature for the TSA was 10–50 °C (10°< TSA < 20°) at depths ranging from (TSA – 10) × 5 km to 60 + (TSA – 10) × 5 km for a flat slab after a subduction history of 7 Myrs. The along-arc slab curvature affects the variation in TSA. The slab radius also appeared to influence the radius of induced mantle flow.

Numerical modeling of three-dimensional convection in the upper mantle of the earth beneath Eurasia lithosphere

We constructed a three-dimensional numerical model of thermal convection in the upper mantle of the Earth in spherical variables using an artificial compressibility method. Results of three-dimensional modeling of convection beneath the Eurasia cratons are presented. The calculation results illustrate the structure of convective flows.

Comments on “Analogies of Ocean/Atmosphere Rotating Fluid Dynamics with Gyroscopes: Teaching Opportunities”

W e welcome the interesting recent paper by Haine and Cherian (2013, hereafter HC13), which introduced a physical analogy between the motion of a mechanical gyroscope and two geophysical flow phenomena that occur in the rotating shallow-water (RSW) equations: namely, geostrophic f low and inertial oscillations. HC13 found, for instance, that geostrophy corresponds to gyroscope precession and that inertial oscillations correspond to gyroscope nutation. They also pointed to other examples of physical analogies, including those between the celebrated Lorenz (1963) low-order model (LOM) of thermal convection and phenomena occurring in lasers, the dynamics of a thermohaline loop oscillator, and El Nino–Southern Oscillation (HC13, p. 682). HC13 did not mention, however, that an earlier LOM of barotropic flow (Lorenz 1960) established an analogy between atmospheric dynamics and gyroscopes (Obukhov and Dolzhansky 1975). Besides, the formal correspondence between the theories of fluid mechanics and of rigid-body mechanics, based on Lie algebras, was revealed by Arnol’d (1989). Finally, systems of equations having the form of several coupled such gyroscopes, as models for fluid flow, had been discussed even earlier by A. N. Kolmogorov during a 1958 seminar in Moscow (see Arnol’d 1991) and by A. M. Obukhov and collaborators in a series of papers beginning with Obukhov (1969). In this Comment we wish to expand the discussion of HC13. Note first that the Lorenz (1963) model was shown (Gluhovsky 1982) to be equivalent to the simplest Volterra gyrostat in a forced, dissipative regime. A gyrostat is a system of bodies whose relative motion does not alter the mass distribution of the system: for example, a gyroscope attached to an axisymmetric rotor; in a Volterra (1899) gyrostat, the rotor spins at a constant angular velocity relative to its “carrier” body (Wittenburg 2008). Second, the Euler gyroscope in the presence of gravity is described by the Euler–Poisson equations (Leimanis 1965). Geostrophic models thereof were considered by Gluhovsky and Dolzhansky (1980) in a study of convection within a rotating ellipsoid under horizontally nonuniform heating. When the gravity acts along the axis of rotation, the ordinary gyroscope (Euler) equations result. When there is an angle between the axis of rotation and the direction of gravity, a three-model chaotic system with three nonlinear terms was obtained, which in the case of a symmetric ellipsoid proved equivalent to the Lorenz (1963) model. Third, the use of LOMs has an established tradition in the study of fluid dynamics and atmospheric flows. Extending the analogy between hydrodynamic LOMs and gyroscopes to include LOMs in the form of coupled gyrostats ensures that energy is conserved in the limit of no dissipation and forcing (Gluhovsky and Tong 1999). This fundamental property of the original partial differential equations is not necessarily retained in general LOMs, which may lead to unphysical behaviors (e.g., Thiffeault and Horton 1996; Gluhovsky and Tong 1999). LOMs for RSW systems (Lorenz 1986; Bokhove and Shepherd 1996) can also be presented as coupled gyrostats, as well as LOMs found in other applications, including shell models of turbulence and Hamiltonian LOMs in geophysical f luid dynamics (Gluhovsky and Tong 1999; Gluhovsky et al. 2002; Gluhovsky 2006; Tong and Gluhovsky 2002, 2008).

Solutal-convection regimes in a two-dimensional porous medium

AbstractWe numerically characterize the temporal regimes for solutal convection from almost first contact to high dissolved solute concentration in a two-dimensional ideal porous layer for Rayleigh numbers $\mathcal{R}$ between $100$ and $5\times 10^4$. The lower boundary is impenetrable. The upper boundary is saturated with dissolved solute and either impermeable or partially permeable to fluid flow. In the impermeable case, initially there is pure diffusion of solute away from the upper boundary, followed by the birth and growth of convective fingers. Eventually fingers interact and merge, generating complex downwelling plumes. Once the inter-plume spacing is sufficient, small protoplumes reinitiate on the boundary layer and are swept into the primary plumes. The flow is now in a universal regime characterized by a constant (dimensionless) dissolution flux $F=0.017$ (the rate at which solute dissolves from the upper boundary). The horizontally averaged concentration profile stretches as a simple self-similar wedge beneath a diffusive horizontal boundary layer. Throughout, the plume width broadens proportionally to $\sqrt{t}$, where $t$ is (dimensionless) time. The above behaviour is parameter independent; the Rayleigh number only controls when transition occurs to a final $\mathcal{R}$-dependent shut-down regime. For the constant-flux and shut-down regimes, we rigourously derive upscaled equations connecting the horizontally averaged concentration, vertical advective flux and plume widths. These are partially complete; a universal expression for the plume width remains elusive. We complement these governing equations with phenomenological boundary conditions based on a marginally stable diffusive boundary layer at the top and zero advective flux at the bottom. Making appropriate approximations in each regime, we find good agreement between predictions from this model and simulated results for both solutal and thermal convection. In the partially permeable upper boundary case, fluid from the convecting layer can penetrate an overlying separate-phase-solute bearing layer where it immediately saturates. The regime diagram remains almost the same as for the impermeable case, but the dissolution flux is significantly augmented. Our work is motivated by dissolution of carbon dioxide relevant to geological storage, and we conclude with a simple flux parameterization for inclusion in gravity current models and suggest that the upscaled equations could lay the foundation for accurate inclusion of dissolution in reservoir simulators.

A Multipoint Initial-Final Value Problem for a Linear Model of Plane-Parallel Thermal Convection in Viscoelastic Incompressible Fluid

S.A. Zagrebina, South Ural State University, Chelyabinsk, Russian Federation, zagrebina_sophiya@mail.ru. Софья Александровна Загребина, кафедра Дифференциальные и стохастические уравнения, Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), zagrebina_sophiya@mail.ru

Predicting surface dynamic topographies of stagnant lid planetary bodies

Although planetary mantles are viscoelastic media, numerical models of thermal convection in a viscoelastic spherical shell are still very challenging. Here, we examine the validity of simplified mechanical and rheological frameworks classically used to approximate viscoelastic dynamic topography. We compare three simplified approaches to a linear Maxwell viscoelastic shell with a pseudo upper free-surface, considered as the reference model. A viscous model with a free-slip boundary condition at the surface correctly reproduces the final relaxed shape of the viscoelastic body but it cannot reproduce the time evolution of the viscoelastic topography. Nevertheless, characterizing the topography development is important since it can represent a significant fraction of the history for planets having a thick and rigid lithosphere (e.g. Mars). A viscous model with a pseudo free-surface, despite its time-dependency, also systematically fails to describe correctly these transient stages. An elastic filtering of the instantaneous viscous topography is required to capture the essence of the time evolution of the topography. We show that a single effective elastic thickness is needed to correctly reproduce the constant transient viscoelastic topography obtained when the lithosphere corresponds to a step-like viscosity variation, while a time-dependence of the effective elastic thickness must be considered to take account of realistic temperature-dependent viscosity variations in the lithosphere. In this case, the appropriate thickness of the elastic shell can be evaluated, at a given instant, with a simple procedure based on the local Maxwell time. Furthermore, if the elastic filtering is performed using the thin elastic shell formulation, an unrealistic degree-dependence of the thickness of the elastic shell is needed to correctly approximate the viscoelastic topography. We show that a model that fully couples a viscous body to an elastic shell of finite thickness estimated using the local Maxwell time gives the best approximation of the viscoelastic deformation, whatever the degree of the load and the time of loading.

Porous convection with local thermal non-equilibrium temperatures and with Cattaneo effects in the solid

There is increasing interest in convection in local thermal non-equilibrium (LTNE) porous media. This is where the solid skeleton and the fluid may have different temperatures. There is also increasing interest in thermal wave motion, especially at the microscale and nanoscale, and particularly in solids. Much of this work has been based on the famous model proposed by Carlo Cattaneo in 1948. In this paper, we develop a model for thermal convection in a fluid-saturated Darcy porous medium allowing the solid and fluid parts to be at different temperatures. However, we base our thermodynamics for the fluid on Fourier's law of heat conduction, whereas we allow the solid skeleton to transfer heat by means of the Cattaneo heat flux theory. This leads to a novel system of partial differential equations involving Darcy's law, a parabolic fluid temperature equation and effectively a hyperbolic solid skeleton temperature equation. This system leads to novel physics, and oscillatory convection is found, whereas for the standard LTNE Darcy model, this does not exist. We are also able to derive a rigorous nonlinear global stability theory, unlike work in thermal convection in other second sound systems in porous media.

Two-dimensional thermal modeling of subduction of the Philippine Sea plate beneath southwest Japan

In this study, we constructed a two-dimensional thermal convection model associated with subduction of the Philippine Sea (PHS) plate beneath southwest Japan to estimate the thermal state. To evaluate the validity of the calculated temperature distribution, we compared the calculated heat flows with observed heat flow data such as the High Sensitivity Seismograph Network Japan (Hi-net) borehole, land borehole, marine probe, and bottom-simulating reflectors. Hi-net heat flow data, which have never been used for thermal modeling, have spatially high resolutions on land and enable detailed estimation of the thermal state beneath the Japanese islands. We considered the spatio-temporal change in the age of the PHS plate, change in its plate motion direction at 3 Ma, and the geometry of its upper surface. Calculated heat flows associated solely with subduction of the PHS plate, passing through central and eastern Shikoku and the Kii Peninsula, were not consistent with observations. They tended to be lower on the seaward side of the corner of the mantle wedge and to have much longer spatial wavelengths on the continental side. To explain heat flow data in southwest Japan, frictional heating at the plate interface on the seaward side of the corner of the mantle wedge and temperature changes due to surface erosion and sedimentation on land associated with crustal deformation during the Quaternary must be incorporated into the model. The most suitable pore pressure ratio to explain the heat flow data is estimated to be 0.95. Temperatures on the upper surface of the PHS plate obtained in this study are lower at shallow depths and higher at greater depths than those obtained by Hyndman et al. (1995) , indicating that the seismogenic zone inferred from this study is narrower than that of their model. Microearthquakes within the subducting PHS plate occur at temperatures below 700–800 °C. • By developing a source code, we calculated the temperature field in southwest Japan. • We used current dense heat flow stations to evaluate thermal structure. • We found that erosion, sedimentation, and frictional heating were required. • The seismogenic zone is narrower than that obtained by Hyndman et al. (1995) . • Intraslab microearthquakes within the PHS plate take place below 700–800 °C.

Differences between tangential geostrophy and columnar flow

SUMMARY Core flows inverted from time-dependent geomagnetic field models image the geodynamo at the top of its generation region, the Earth’s outer core. Physical assumptions incorporated in theseinversionsaffecttheresultingflows.Basedonrapidrotationdominance,twoassumptions similar in form yet different in essence have been proposed: tangential geostrophy (TG, LeMou¨ 1984) and columnarflow (CF, Amit & Olson 2004). We recall that CF is theoretically consistent with the quasi-geostrophy (QG) theory for an incompressible fluid with spherical solid boundaries whereas TG is not. As such, we highlight the importance of applying the CF assumptionwheninvertinggeomagneticdataforinteriorcore(columnar)flowsthatcanbeused in kinematic dynamo and thermal convection models in the Boussinesq approximation. Next we evaluate the non-uniqueness associated with CF flows. The areas of ambiguous patches at the core surface where invisible TG or CF flows reside are roughly comparable. The spatial distribution of ambiguous patches for both TG and CF is quite asymmetric about the equator, so assuming equatorial symmetry is expected to reduce the non-uniqueness significantly. In fact, for assumed equatorial symmetry, the only possible non-unique flows will be those along hypothetical ζ-contours in the opposite hemispheres that their equatorial plane projections are parallel. TG flows exhibit a strong Atlantic/Pacific hemispheric dichotomy and a well-defined eccentric gyre whereas in CF flows the dichotomy between these two hemispheres is weaker and the gyre is less clear suggesting that the eccentric gyre might not conserve mass. Both TG and CF upwelling/downwelling patterns are strongly localized in the equatorial region. In addition, in both cases upwelling/downwelling is correlated with equatorward/poleward flow respectively, as expected for QG convection. CF upwelling is more intense than TG upwelling but the magnitude ratio is smaller than the factor 2 distinguishing the analytical expressions of the two assumptions. This smaller magnitude ratio is due to the fact that presently observed geomagnetic secular variation features are mostly explained by magnetic field advection by toroidal core flow in the frozen-flux approximation. Robust upwelling features below India/Indonesia may be viewed as geomagnetic evidence for whole core convection.

Mantle dynamics with pressure- and temperature-dependent thermal expansivity and conductivity

In numerical simulations of mantle convection it is commonly assumed that the coefficients of thermal expansion α and thermal conduction k are either constant or pressure-dependent. Pressure changes are generally computed using parametrizations that rely on extrapolations of low-pressure data for a single upper-mantle phase. Here we collect data for both the pressure and temperature dependence of α from a database of first-principles calculations, and of k from recent experimental studies. We use these data-sets to construct analytical parametrizations of α and k for the major upper- and lower-mantle phases that can be easily incorporated into exisiting convection codes. We then analyze the impact of such parametrizations on Earth’s mantle dynamics by employing two-dimensional numerical models of thermal convection. When α is the only variable parameter, both its temperature and pressure dependence enhance hot plumes and tend to inhibit the descent of cold downwellings. Taking into account a variable k leads to a strong increase of the bulk mantle temperature, which reduces the buoyancy available to amplify bottom boundary layer instabilities and causes mantle flow to be driven primarily by the instability of cold plates whose surface velocity also tends to rise. When both parameters are considered together, we observe an increased propensity to local layering which favors slab stagnation in the transition zone and subsequent thickening in the lower mantle. Furthermore, the values of k near the core-mantle boundary ultimately control the effect of this physical property on convection, which stresses the importance of determining the thermal conductivity of the post-perovskite phase.

Triply resonant penetrative convection

A thermal convection model is considered that consists of a layer of viscous incompressible fluid contained between two horizontal planes. Gravity is acting vertically downward, and the fluid has a density maximum in the active temperature range. A heat source/sink that varies with vertical height is imposed. It is shown that in this situation there are three possible (different) sub-layers that may induce convective overturning instability. The possibility of resonance between the motion in these layers is investigated. A region is discovered where a very sharp increase in Rayleigh number is observed. In addition to a linearized instability analysis, two global (unconditional) nonlinear stability thresholds are derived.

Investigation and modeling of thermal air convection and transport of local precipitation during winter operation of the service spillway at the Sayano-Shushenskaya HPP

After the failure at the Sayano-Shushenskaya HPP, the need has arisen for prediction of winter operating conditions of the service spillway, since, it had previously not been placed in service at negative ambient temperatures. . For this purpose, coworkers of the JSC NIIES and the Computer Center at the Russian Academy of Sciences conducted a scientific-research study during the fall and winter of 2009 – 2010 on the subject “Prediction of the effect of the water-ice cloud in the air beyond the spillway of the Sayano-Shushenskaya HPP during the winter,” some results of which are presented in this paper. As we know, passage of water at the Sayano-Shushenskaya HPP with the generating sets idle occurred during the winter of 20092010 only over the service spillway, all 11 bays of which were operated at the minimum opening (half stage). Despite the fact that operation at half stage provided minimal aeration of the flow, and, as a consequence, minimum generation of water droplets in the toe basin and minimal transport of droplets and frozen mist to the crest of the dam by ascending air flows, this, everything being equal, led to the two following negative consequences: — rapid formation of thick ice layers on the faces of the spillway chutes and platforms adjacent to the toe basin (for example, on the crane trestle of the spillway dam (Fig. 1), and on the section of the separating abutment near the spillway; and, — formation and transport (Fig. 2) of solid precipitation (ice grit, snow), which are deposited on the more remote platforms (for example, on the balcony above the open section of the spillway, the crest of the dam, the roof of the machine room, etc.). These effects were observed in the period when the mean-daily air temperature in the region of the Sayano-Shushenskaya HPP had been consistently negative. The objective of this study was to evaluate by computational means (using both empirical relationships, and also mathematical modeling) the rate of ice formation and precipitation (liquid and solid) on platforms of the HPP (Fig. 1), for example, at the level of crane trestle of the spillway, where direct measurements were impossible.

Time-dependent convection models of mantle thermal structure constrained by seismic tomography and geodynamics: implications for mantle plume dynamics and CMB heat flux

SUMMARY One of the outstanding problems in modern geodynamics is the development of thermal convection models that are consistent with the present-day flow dynamics in the Earth’s mantle, in accord with seismic tomographic images of 3-D Earth structure, and that are also capable of providing a time-dependent evolution of the mantle thermal structure that is as ‘realistic’ (Earth-like) as possible. A successful realization of this objective would provide a realistic model of 3-D mantle convection that has optimal consistency with a wide suite of seismic, geodynamic and mineral physical constraints on mantle structure and thermodynamic properties. To address this challenge, we have constructed a time-dependent, compressible convection model in 3-D spherical geometry that is consistent with tomography-based instantaneous flow dynamics, using an updated and revised pseudo-spectral numerical method. The novel feature of our numerical solutions is that the equations of conservation of mass and momentum are solved only once in terms of spectral Green’s functions. We initially focus on the theory and numerical methods employed to solve the equation of thermal energy conservation using the Green’s function solutions for the equation of motion, with special attention placed on the numerical accuracy and stability of the convection solutions. A particular concern is the verification of the global energy balance in the dissipative, compressible-mantle formulation we adopt. Such validation is essential because we then present geodynamically constrained convection solutions over billion-year timescales, starting from present-day seismically constrained thermal images of the mantle. The use of geodynamically constrained spectral Green’s functions facilitates the modelling of the dynamic impact on the mantle evolution of: (1) depth-dependent thermal conductivity profiles, (2) extreme variations of viscosity over depth and (3) different surface boundary conditions, in this case mobile surface plates and a rigid surface. The thermal interpretation of seismic tomography models does not provide a radial profile of the horizontally averaged temperature (i.e. the geotherm) in the mantle. One important goal of this study is to obtain a steady-state geotherm with boundary layers which satisfies energy balance of the system and provides the starting point for more realistic numerical simulations of the Earth’s evolution. We obtain surface heat flux in the range of Earth-like values : 37 TW for a rigid surface and 44 TW for a surface with tectonic plates coupled to the mantle flow. Also, our convection simulations deliver CMB heat flux that is on the high end of previously estimated values, namely 13 TW and 20 TW, for rigid and plate-like surface boundary conditions, respectively. We finally employ these two end-member surface boundary conditions to explore the very-long-time scale evolution of convection over billion-year time windows. These billion-year-scale simulations will allow us to determine the extent to which a ‘memory’ of the starting tomography-based thermal structure is preserved and hence to explore the longevity of the structures in the present-day mantle. The two surface boundary conditions, along with the geodynamically inferred radial viscosity profiles, yield steady-state convective flows that are dominated by long wavelengths throughout the lower mantle. The rigid-surface condition yields a spectrum of mantle heterogeneity dominated by spherical harmonic degree 3 and 4, and the plate-like surface condition yields a pattern dominated by degree 1. Our exploration of the time-dependence of the spatial heterogeneity shows that, for both types of surface boundary condition, deep-mantle hot upwellings resolved in the present-day tomography model are durable and stable features. These deeply rooted mantle plumes show remarkable longevity over very long geological time spans, mainly owing to the geodynamically inferred high viscosity in the lower mantle.