Initial Mantle Temperature Research Articles

Scaling laws for stagnant-lid convection with a buoyant crust

SUMMARY Stagnant-lid convection, where subduction and surface plate motion is absent, is common among the rocky planets and moons in our solar system, and likely among rocky exoplanets as well. How stagnant-lid planets thermally evolve is an important issue, dictating not just their interior evolution but also the evolution of their atmospheres via volcanic degassing. On stagnant-lid planets, the crust is not recycled by subduction and can potentially grow thick enough to significantly impact convection beneath the stagnant lid. We perform numerical models of stagnant-lid convection to determine new scaling laws for convective heat flux that specifically account for the presence of a buoyant crustal layer. We systematically vary the crustal layer thickness, crustal layer density, Rayleigh number and Frank–Kamenetskii parameter for viscosity to map out system behaviour and determine the new scaling laws. We find two end-member regimes of behaviour: a ‘thin crust limit’, where convection is largely unaffected by the presence of the crust, and the thickness of the lithosphere is approximately the same as it would be if the crust were absent; and a ‘thick crust limit’, where the crustal thickness itself determines the lithospheric thickness and heat flux. Scaling laws for both limits are developed and fit the numerical model results well. Applying these scaling laws to rocky stagnant-lid planets, we find that the crustal thickness needed for convection to enter the thick crust limit decreases with increasing mantle temperature and decreasing mantle reference viscosity. Moreover, if crustal thickness is limited by the formation of dense eclogite, and foundering of this dense lower crust, then smaller planets are more likely to enter the thick crust limit because their crusts can grow thicker before reaching the pressure where eclogite forms. When convection is in the thick crust limit, mantle heat flux is suppressed. As a result, mantle temperatures can be elevated by 100 s of degrees K for up to a few Gyr in comparison to a planet with a thin crust. Whether convection enters the thick crust limit during a planet’s thermal evolution also depends on the initial mantle temperature, so a thick, buoyant crust additionally acts to preserve the influence of initial conditions on stagnant-lid planets for far longer than previous thermal evolution models, which ignore the effects of a thick crust, have found.

Toward Constraining Mars' Thermal Evolution Using Machine Learning

AbstractThe thermal and convective evolution of terrestrial planets like Mars is governed by a number of initial conditions and parameters, which are poorly constrained. We use Mixture Density Networks (MDN) to invert various sets of synthetic present‐day observables and infer five parameters: reference viscosity, activation energy and activation volume of the diffusion creep rheology, an enrichment factor for radiogenic elements in the crust, and initial mantle temperature. The data set comes from 6,130 two‐dimensional simulations of the thermal evolution of Mars' interior. We quantify the possibility of constraining a parameter using the log‐likelihood value from the MDN. Reference viscosity can be constrained to within 32% of its entire range (1019 − 1022 Pa s), when all the observables are available: core‐mantle‐boundary heat flux, surface heat flux, radial contraction, melt produced, and duration of volcanism. Furthermore, crustal enrichment factor (1–50) can be constrained, at best, to within 15%, and the activation energy (105 − 5 × 105 J mol−1) to within 80%. Initial mantle temperature can be constrained to within 39% of its range (1,600–1,800 K). Using the full present‐day temperature profile or parts of it as an observable tightens the constraints further. The activation volume (4 × 10−6 − 10 × 10−6 m3 mol−1) cannot be constrained. We also tested different levels of uncertainty in the observables and found that constraints on different parameters loosen differently, with initial temperature being the most sensitive. Finally, we present how a joint probability model for all parameters can be obtained from the MDN.

Carbon cycling and interior evolution of water-covered plate tectonics and stagnant-lid planets

Aims. The long-term carbon cycle for planets with a surface entirely covered by oceans works differently from that of the present-day Earth because inefficient erosion leads to a strong dependence of the weathering rate on the rate of volcanism. In this paper, we investigate the long-term carbon cycle for these planets throughout their evolution. Methods. We built box models of the long-term carbon cycle based on CO2 degassing, seafloor-weathering, metamorphic decarbonation, and ingassing and coupled them with thermal evolution models of plate tectonics and stagnant-lid planets. Results. The assumed relationship between the seafloor-weathering rate and the atmospheric CO2 or the surface temperature strongly influences the climate evolution for both tectonic regimes. For a planet with plate tectonics, the atmospheric CO2 partial pressure is characterized by an equilibrium between ingassing and degassing and depends on the temperature gradient in subduction zones affecting the stability of carbonates. For a stagnant lid planet, partial melting and degassing are always accompanied by decarbonation, such that the combined carbon content of the crust and atmosphere increases with time. While the initial mantle temperature on planets with plate tectonics only affects the early evolution, it influences the evolution of the surface temperature of stagnant-lid planets for much longer. Conclusions. For both tectonic regimes, mantle cooling results in a decreasing atmospheric CO2 partial pressure. For a planet with plate tectonics this is caused by an increasing fraction of subduction zones that avoid crustal decarbonation, and for stagnant-lid planets this is caused by an increasing decarbonation depth. This mechanism may partly compensate for the increase of the surface temperature due to increasing solar luminosity with time, and thereby contribute to keeping planets habitable in the long-term.

Using pattern recognition to infer parameters governing mantle convection

The results of mantle convection simulations are fully determined by the input parameters and boundary conditions used. These input parameters can be for initialisation, such as initial mantle temperature, or can be constant values, such as viscosity exponents. However, knowledge of Earth-like values for many input parameters are very poorly constrained, introducing large uncertainties into the simulation of mantle flow. Convection is highly non-linear, therefore linearised inversion methods cannot be used to recover past configurations over more than very short periods of time, which makes finding both initial and constant simulation input parameters very difficult. In this paper, we demonstrate a new method for making inferences about simulation input parameters from observations of the mantle temperature field after billions of years of convection. The method is fully probabilistic. We use prior sampling to construct probability density functions for convection simulation input parameters, which are represented using neural networks. Assuming smoothness, we need relatively few samples to make inferences, making this approach much more computationally tractable than other probabilistic inversion methods. As a proof of concept, we show that our method can invert the amplitude spectra of temperature fields from 2D convection simulations, to constrain yield stress, surface reference viscosity and the initial thickness of primordial material at the CMB, for our synthetic test cases. The best constrained parameter is yield stress. The reference viscosity and initial thickness of primordial material can also be inferred reasonably well after several billion years of convection.

Influence of magmatism on mantle cooling, surface heat flow and Urey ratio

Two-dimensional thermo-chemical mantle convection simulations are used to investigate the influence of melting-inducted differentiation on the thermal evolution of Earth's mantle, focussing in particular on matching the present-day surface heat flow and the ‘Urey ratio’. The influence of internal heating rate, initial mantle temperature and partitioning of heat-producing elements into basaltic crust are studied. High initial mantle temperatures, which are expected following Earth's accretion, cause major differences in early mantle thermo-chemical structures, but by the present-day surface heat flux and internal structures are indistinguishable from cases with a low initial temperature. Assuming three different values of mantle heat production that vary by more than a factor of two results in small differences in present-day heat flow, as does assuming different partitioning ratios of heat-producing elements into crust. Indeed, all of the cases presented here, regardless of exact parameters, have approximately Earth's present-day heat flow, with substantial fractions coming from the core and from mantle cooling. As a consequence of the model present-day surface heat flow varying only slightly with parameters, the Urey ratio (the ratio of total heat production to the total surface heat flow) is highly dependent on the amount of internal heat production, and due to the large uncertainty in this, the Urey ratio is considered to be a much poorer constraint on thermal evolution than the heat flow. The range of present-day Urey ratio observed in simulations here is about 0.3 to 0.5, which is consistent with observational and geochemical constraints (Jaupart et al., 2007). Magmatic heat transport contributes an upper bound of 9% to Earth's present-day heat loss but a much higher fraction at earlier times—often more than convective heat loss—so neglecting this causes an overestimation of the Urey ratio. Magmatic heat transport also plays an important role in mantle cooling. Considering these points, it is important to include magmatic effects when attempting to understand the thermal evolution of the Earth.

Numerical models of Martian mantle evolution induced by magmatism and solid-state convection beneath stagnant lithosphere

[1] Numerical models of magmatism in a convecting mantle are presented to understand Martian mantle evolution. The viscosity is strongly temperature dependent, and the lithosphere is stagnant. Magmatism is modeled as a permeable flow of basaltic magma generated by decompression melting. When the initial mantle temperature is sufficiently high, a magma ocean develops to generate a thick basaltic crust and create a compositionally layered mantle. The upper layer of the mantle consists of compositionally buoyant residue of the basaltic crust, while the lower layer consists of compositionally denser materials that are not as depleted in the basaltic component. Hot plumes grow from the lower layer making it thinner with time by erosion. The hot plumes also induce magmatism, which extracts heat-producing elements (HPEs) from the mantle. The HPE extraction regulates plume magmatism activity and controls the duration of plume magmatism. Plume magmatism also keeps the mantle temperature below the solidus by efficiently extracting heat as soon as the mantle temperature exceeds the solidus. The Martian mantle is likely to have evolved as a relaxation from a compositionally layered state formed by the magma ocean, and plume magmatism has probably played a crucial role in the relaxation process.

New constraints on the thermal and volatile evolution of Mars

The thermal and volatile evolution of Mars has not been studied from the perspective of consistency with the preservation of the Martian global dichotomy and with estimates of the elastic thickness over time. We use three thermal evolution models for Mars: (1) stagnant lid, (2) early plate tectonics followed by stagnant lid, and (3) mantle overturn, to calculate the amount of relaxation of the dichotomy boundary and elastic thickness values for Noachian- and Hesperian-aged terrains. To explore a wide range of parameters, we evaluate two different initial mantle temperatures, and wet and dry rheologies. Our model results show that the relative water content of the crust has an effect roughly equal to 500 K variations of initial mantle temperature. For all three thermal models, a lower crust viscosity of 10 20–10 21 Pa s during the first 0.1 Ga after formation of dichotomy would allow for the preservation of the long-wavelength topography of Mars and fitting of the elastic thickness. This viscosity range implies either wet, cold (∼1500 K) lower crust, or dry, hot (∼2000 K) lower crust in Noachian. Additional constraints are necessary to distinguish between the individual thermal models. For the stagnant lid model, neither the cold, wet crust nor the hot, dry crust agree with timing and amount of the crustal production [Hauck II., S.A., Phillips, R.J., 2002. Thermal and crustal evolution of Mars. J. Geophys. Res. 107, 5052]. Moreover, drying of the crust is required for this model in order to match the admittance elastic thickness at the Hesperian/Amazonian boundary implying remelting of the crust. The hot, dry crust in the early plate tectonics model limits a plate tectonic epoch to only 100–200 Myr and implies dry mantle, which is in disagreement with water found in meteorites. The cold, wet crustal rheology implies the formation of crust during the plate tectonics regime because of the low crustal production during the stagnant lid regime. For mantle overturn, the temperature required for wet crust does not fit the original mantle profile while the dry crust does; however, in order to explain the initially hot thermal profile the crust must have been emplaced very fast. Generally, dry crustal rheology does not fit low elastic values in the Hesperian and implies either that rheology may differ between the southern and northern hemispheres: wet in northern hemisphere and dry in southern hemisphere, or that local weakening occurred. Wet crustal rheology fits well all elastic data except S. Hellas rim, which may be anomalous. Mantle rheology is unconstrained by our modeling and can be either dry or wet.

Viscosity of the Martian mantle and its initial temperature: Constraints from crust formation history and the evolution of the magnetic field

The rheology of the Martian mantle and the planet's initial temperature is constrained with thermal evolution models that include crust growth and test the conditions for magnetic field generation in the core. As observations we use the present-day average crustal thickness of 50–120 km as estimated from the Mars Global Surveyor gravity and topography data, the evidence for the crust being produced mostly early, with a rate declining from the Noachian to the Hesperian, and the evidence for an early magnetic field that likely existed for less than a billion years. We use the fact that the rate of crust growth is a function of temperature, which must be above the solidus in the sub-lithosphere mantle, and the mantle convection speed because the latter determines the rate at which melt can be replenished. The convection speed is a strong function of viscosity which, in turn, is a strong function of temperature and also of the water content of the mantle. We use a viscosity parameterization with a reference viscosity evaluated at 1600 K the value of which can be characteristic of either a dry or a wet mantle. We further consider the Fe–FeS phase diagram for the core and compare the core liquidus estimated for a sulphur content of 14% as suggested by the SNC meteorite compositions with the core temperatures calculated for our cooling models. Two data sets of the Fe–FeS eutectic temperature have been used that differ by about 200 K [Böhler, R., 1996. Fe–FeS eutectic temperatures at 620 kbar. Phys. Earth Planet. Inter. 96, 181–186; Fei, Y., Bertka, C.M., Finger, L.W., 1997. High-pressure iron–sulphur compound, Fe 3S 2, and melting relations in the Fe–FeS system. Science 275, 1621–1623] at Martian core–mantle boundary pressure and in the eutectic composition by 5 wt%. The differences in eutectic temperature and composition translate into a difference of about 400 K in liquidus temperature for 14 wt% sulphur. We find it premature to rule out specific mantle rheologies on the basis of the presently available crustal thickness and crust growth evidence. Rather a trade-off exists between the initial mantle temperature and the reference viscosity. Both a wet mantle rheology with a reference viscosity less than 10 20 Pas and a dry mantle rheology with a reference viscosity of 10 21 Pas or more can be acceptable if initial mantle temperatures between roughly 1700 and 2000 K are allowed. To explain the magnetic field history, the differences in liquidus temperatures matter. For a liquidus temperature of about 1900 K at the Martian core–mantle boundary as calculated from the Böhler et al. eutectic, a dry mantle rheology can best explain the lack of a present-day dynamo. For a liquidus temperature of about 1500 K at the core–mantle boundary as calculated from the Fei et al. eutectic all models are consistent with the observed lack of dynamo action. The reason lies with the fact that at 14 wt% S the Martian core would be close to the eutectic composition if the Fei et al. data are correct. As inner core growth is unlikely for an almost eutectic core, the early field would have been generated by a thermally driven dynamo. Together with the measured strength of the Martian crustal magnetization this would prove the feasibility of a strong thermally driven dynamo.

Evolution of the global mean heat flow over 4.6 Gyr

The mean global heat flow of a planet depends mainly on the initial mantle temperature, the distribution of the radiogenic heat sources, and the mechanism of the heat transport within the mantle. It is well known that subsolidus convection is the dominant mechanism of the heat transport within the mantle. With the help of parametrized convection models (Franck and Bounama, 1995; and references given therein) temporal variations of the average mantle temperature and the surface heat flow in terms of the Rayleigh number can be calculated under certain initial conditions. In our numerical experiments we calculate the thermal and the degassing history for two different initial average mantle temperatures: 2400 K and 3000 K. Looking at the two different starting temperatures, we find qualitatively different behaviour in the first 1–1.5 Gyr of thermal evolution. After this stage the two curves converge rapidly showing the so-called readjustment-effect. After readjustment the quantities decrease monotonically. Furthermore, we investigate the influence of continental growth and the thermal and degassing history by using simple analytical continental growth models. We compare our results with that of laboratory melting experiments of komatiites providing estimates for the secular mantle cooling and discuss relations between global mean heat flow and the areal spreading rate.