Mantle convection is the principal control on the thermal and geological evolution of the earth, including the motion of the tectonic plates, which in turn influences earthquakes, tsunamis, and volcanic eruptions. This system is governed by the equations for balance of mass, momentum, and energy for a viscous incompressible non-Newtonian fluid. Taking present-day temperatures as given, the time dependence can be neglected, eliminating the energy equation. In this case, the physics of the mantle are modeled by the Stokes equation with nonlinear rheology (the so-called forward problem). This dissertation focuses on solving the mantle convection inverse problem governed by the nonlinear Stokes forward problem with full nonlinear rheology, with an infinite-dimensional adjoint-based inversion method. The need for inverse methods in the study of mantle convection stems from the fact that the constitutive parameters are subject to uncertainty. Inversion for nonlinear rheology parameters presents considerable difficulties, which are explored in this dissertation. A spectral analysis of the Hessian operator is performed to investigate the ill-posedness of the inverse problem. The general form of the numerical eigenvalues is found to agree with that of the theoretically-derived ones (based on a model 1D Stokes problem), both of which collapse rapidly to zero, suggesting a high degree of ill-posedness. This motivates the use in this thesis of regularizations that are of Tikhonov type (favoring smooth viscosity) and total variation type (favoring piecewise-smooth viscosity). In addition, the eigenfunctions of the Hessian indicate that increasingly smaller length scales of viscosity are increasingly less observable, and that resolution decays with depth. The wide range of spatial scales of interest (varying from 1 km scale associated with plate boundaries to 10⁴ km global scales) prompts the use of adaptive mesh refinement in a parallel framework. The results show that both higher levels of nonlinearity and larger orders of magnitude of variation in the viscosity cause the inverse problem to be more ill-conditioned, increasing the difficulty of solving the inverse problem. Despite the severe ill-posedness of the inverse problem, stemming from the small number of observations compared to large number of degrees of freedom of the viscosity parameters, with the correct regularization weight and the right type of regularization, it is possible to reasonably infer information about the viscosity of the mantle, particularly in shallow regions. A number of 2D and 3D inversions are shown to demonstrate these capabilities. / text

ASPECT is a new convection code that uses more modern and advanced solver methods than geodynamics legacy codes. I use ASPECT to calculate 2-dimensional Cartesian as well as 2- and 3-dimensional spherical-shell convection cases. All cases use the Boussinesq approximation. The 2D cases come from Blankenbach et al. (1989), van Keken et al. (1997), and Davies et al. (in preparation). Results for 2D cases agree well with their respective benchmark papers. The time-evolutions of the root mean square velocity (Vrms) and Nusselt number agree, often to within 1%. The 3D cases come from Zhong et al. (2008). Modifications were made to the simple.cc and harmonic_perturbation.cc files in the ASPECT code in order to reproduce the initial conditions and temperature-dependence of the rheology used in the benchmark. Cases are compared using both CitcomS and ASPECT with different levels of grid spacing, as well as comparing uniform grid spacing and the ASPECT default grid spacing, which refines toward the center. Results for Vrms, average temperature, and Nusselt numbers at the top and bottom of the shell range from better than 1% agreement between CitcomS and ASPECT for cases with tetragonal planforms and 7000 Rayleigh number to as much as 44% difference for cases with cubic planforms and 10^5 Rayleigh number. For all benchmarks, the top Nusselt number from ASPECT is farthest from the reported benchmark values. The 3D planform and radially averaged quantity plots agree. I present these results, as well as recommendations and possible fixes for discrepancies in the results, specifically in the Nusselt numbers, Vrms, and average temperature. / Master of Science

One of the defining characteristics of hotspot volcanism is the presence of a long-lived, linear chain of age-progressive volcanoes created by the movement of the lithosphere over a stationary melting anomaly. However, the spatial distribution of volcanism at hotspots is often complex and highly variable suggesting that the relationship between magma generation and magma transport at hotspots is poorly understood. Here, I present the results of the first systematic quantitative characterization of the spatial distribution of volcanism at oceanic hotspots. In the first study I develop a novel methodology to characterize the across-strike distribution of volcanism at hotspots and apply it to a catalog of 40 oceanic hotpots. I find that only 25% (10/40) of hotspots exhibit the simple single-peak profile predicted by geodynamic models of melt generation in mantle plumes. The remaining 75% (30/40) of hotspots exhibit a dual- or multi-peak pattern. In the second study, I focus on the across-strike distribution of volcanism at the oceanic hotspots that are sourced by a deep-rooted mantle plume. 14 out of the 15 consensus plume-fed hotspots exhibit a dual-peaked across-strike profile. The spacing between these peaks display a strong negative correlation with lithospheric age, in direct contrast to models of inter-volcanic spacing controlled by elastic plate thickness. This relation suggests a different mechanism controls volcanic spacing at plume-fed hotspots. In the third chapter, I investigate variations in the average topographic profiles over time along the two longest and best-constrained oceanic hotspot tracks: Hawaii and Louisville. I find that the dual-peak across-strike profile of volcanism is a persistent feature at the Louisville hotspot over the entire length of the track examined (spanning a period of more than 65 Myr). In contrast, the dual-peak profile of volcanism at Hawaii is only evident along the most recent portion of the track (i.e., over the last 5 Myr). In total, this thesis represents a significant step foreword in the collective understanding of hotspot volcanism, and introduces a new diagnostic tool for analysis of hotspot influenced seafloor topography.

Through geodynamic modeling, I investigate how Mars could have produced the extensive volcanism required to form the Tharsis rise early in its history, as well as continue to produce small amounts of melt up to present-day, in order to account for the evidence of limited geologically recent volcanism. InSight is the first interplanetary mission dedicated primarily to the study of a planet's deep interior, and has provided useful constraints for the present structure and interior temperature of Mars. I use the results from InSight and other spacecraft missions to more accurately model Mars, and evaluate the results of my geodynamic models, so as to constrain the properties that are necessary for or consistent with both the InSight results and the volcanic history reflected on the surface. This modeling has required extensive modification to the CitcomS geodynamic code I use, the bulk of that effort being in implementing and testing the melting calculations. One of the useful constraints that would have been provided by InSight would have been ground truthing the heat flow from the interior at the landing site, and this required determining, among other quantities, the thermal conductivity of the regolith into which the heat flow probe (mole) was placed. While the mole could not penetrate to its designed depth, thus disallowing the complete heat flow measurement, the team were able to obtain the necessary data determine the thermal conductivity, and how it varies seasonally. My rapid analytical method of estimating thermal conductivity produces results that agree surprisingly well with those of the team's complex numerical model, despite the mole not meeting the assumption of a sufficiently high length to width ratio. / Doctor of Philosophy / I investigate how Mars could have produced the extensive volcanism required to form the Tharsis rise early in its history, as well as continue to produce small amounts of melt up to present-day, in order to account for the evidence of limited geologically recent volcanism. I use 3D computer models of the mantle--the solid, but slowly flowing layer that makes up the bulk of rocky planets like Earth and Mars. InSight is the first interplanetary mission dedicated to the study of a planet's deep interior, and has provided useful constraints for the present structure and interior temperature of Mars. I use the results from InSight and other spacecraft missions to more accurately model Mars, and evaluate the results of my models, so as to constrain the properties that are necessary for or consistent with both the InSight results and the volcanic history reflected on the surface. This modeling has required extensive modification to the modeling code I use, the bulk of that effort being in implementing and testing the melting calculations. One of the useful constraints that would have been provided by InSight would have been ground truthing the heat flow from the interior at the landing site, and this required determining, among other quantities, the thermal conductivity of the soil into which the heat flow probe (mole) was placed. While the mole could not penetrate to its designed depth, thus disallowing the complete heat flow measurement, the team were able to obtain the necessary data determine the thermal conductivity, and how it varies seasonally. My rapid analytical method of estimating thermal conductivity produces results that agree surprisingly well with those of the team's complex numerical model, despite the mole not meeting the assumption of a sufficiently high length to width ratio.

In the current study we developed an internal state variable (ISV) model based on the Bammann inelasticity internal state variable model (BIISV) to include damage, recrystallization, and texture development, which we then implemented into a mantle convection code, TERRA2D, to incorporate higher fidelity material behavior into mantle convection simulations. With experimental stress strain data found in the literature model constants for the BIISV model were determined for a number of geologic materials. The BIISV model was shown to be far superior to the steady state power law model currently used by the geologic community to capture the deformation of geologic materials. Once implemented and verified in TERRA2D the BIISV model revealed locations of hardened material that behaved like diverters in the cold thermal boundary layer that the power law model could never produce. These hardened regions could be a plausible reason for the current subduction zones present on the earth. We then altered the BIISV model equation to include the effects of damage, recrystallization, and texture development in order to model possible weakening mechanisms in the cold thermal boundary layer of the mantle. Inclusion of damage and recrystallization allowed the cold thermal boundary layer to mobilize and plunge downward into the hotter region below. Texture development increased the intensity of rotational flow within the hotter zone as cold boundary material plunged downward which aided in destabilizing the cold upper thermal boundary layer. The inclusion of an internal state variable model with damage, recrystallization, and texture development represents a significant advancement in handling deformational physics for mantle phenomena in a comprehensive, unified, and automatic manner.

O presente trabalho tem como objetivo estudar a dinâmica do manto superior em margens continentais através do uso de modelos numéricos que simulam processos convectivos e condutivos no tempo geológico, avaliando-se como a incorporação da convecçãao no estudo da história de subsidência de margens continentais difere do resultado obtido através de modelos puramente condutivos. Como primeiros testes, foram realizadas comparações dos resultados numéricos com soluções analíticas para diferentes valores do número de Rayleigh, verificando-se a validade das soluções computacionais. Também foi feita uma análise da estabilidade da litosfera no tempo geológico para diferentes perfis de viscosidade, servindo como base para a escolha dos parâmetros reológicos do manto para os modelos no contexto de margens divergentes. A partir dos cenários numéricos que melhor reproduziram a estrutura da litosfera terrestre, novos cenários foram criados para simular a evolução térmica e isostática de margens continentais. Como exemplo, utilizou-se dados geofísicos e geológicos extraídos da literatura para a bacia sedimentar do Golfo do Leão, no sudeste da Françaa, com o objetivo de comparar a evolução geodinâmica do presente modelo numérico com outros modelos publicados na literatura. Observou-se que o efeito convectivo astenosférico preserva a estrutura térmica aquecida da margem estirada por mais tempo em comparação com o modelo puramente condutivo. Isso implica que, possivelmente, outros fatores também devem ser levados em consideração como o efeito da geometria tridimensional da margem do Golfo do Leão que pode contribuir para um aumento da subsidência da margem em relação ao modelo obtido no presente trabalho. Adicionalmente, constatou-se que a convecção mantélica pode induzir tensões na base da litosfera que a deslocam dinamicamente ao longo do tempo geológico, podendo influenciar a evolução estratigráfica das bacias sedimentares marginais. São apresentados cerca de 60 cenários geodinâmicos mostrando como a variação da estrutura reológica do manto influencia a evolução térmica da litosfera e consequentemente, a história de subsidência da margem. / This work aims to study the dynamics of the upper mantle in continental margins by using numerical models that simulate convective and conductive processes in geological time scale. It was evaluated the contribution of convection and conduction for subsidence history of sedimentary basins. As first tests, simple numerical scenarios with different Rayleigh number were compared with analytic solutions, verifying the validate of the computational solutions. These numerical experiments were followed by the analysis of the lithospheric stability in the geological time scale for different values of viscosity. These experiments were used as a base for the choice of the rheological parameters of the mantle for the models in the context of divergent margins. From the numerical scenarios that better reproduced the lithospheric structure of the Earth, new scenarios were created to simulate the thermal and isostatic evolution of continental margins. As an example, geophysical and geological data extracted from the literature for the sedimentary basin of the Gulf of Lion, Southeastern France, were compared with the results of different geodynamic models published in the literature and with the numerical scenarios obtained in the present work. We observed that the effect of the astenospheric convection preserves the thermal structure of the stretched margin for a long time in comparison with purely conductive models. This implies that, possibly, other processes must be taken into account, such as the effect of the three-dimensional geometry of the Gulf of Lion margin that may contribute to a higher subsidence of the margin than the one obtained in the present work. Additionally, it was observed that mantle convection may induce stress at the base of the lithosphere that dynamically moves it in the geological time, and may influence the stratigraphic evolution of sedimentary basins. It is presented about 60 scenarios showing how the variation of the rheological structure of the mantle is taken into account in the thermal evolution of the lithosphere and consequently in the subsidence history of the margin.

O presente trabalho tem como objetivo estudar a dinâmica do manto superior em margens continentais através do uso de modelos numéricos que simulam processos convectivos e condutivos no tempo geológico, avaliando-se como a incorporação da convecçãao no estudo da história de subsidência de margens continentais difere do resultado obtido através de modelos puramente condutivos. Como primeiros testes, foram realizadas comparações dos resultados numéricos com soluções analíticas para diferentes valores do número de Rayleigh, verificando-se a validade das soluções computacionais. Também foi feita uma análise da estabilidade da litosfera no tempo geológico para diferentes perfis de viscosidade, servindo como base para a escolha dos parâmetros reológicos do manto para os modelos no contexto de margens divergentes. A partir dos cenários numéricos que melhor reproduziram a estrutura da litosfera terrestre, novos cenários foram criados para simular a evolução térmica e isostática de margens continentais. Como exemplo, utilizou-se dados geofísicos e geológicos extraídos da literatura para a bacia sedimentar do Golfo do Leão, no sudeste da Françaa, com o objetivo de comparar a evolução geodinâmica do presente modelo numérico com outros modelos publicados na literatura. Observou-se que o efeito convectivo astenosférico preserva a estrutura térmica aquecida da margem estirada por mais tempo em comparação com o modelo puramente condutivo. Isso implica que, possivelmente, outros fatores também devem ser levados em consideração como o efeito da geometria tridimensional da margem do Golfo do Leão que pode contribuir para um aumento da subsidência da margem em relação ao modelo obtido no presente trabalho. Adicionalmente, constatou-se que a convecção mantélica pode induzir tensões na base da litosfera que a deslocam dinamicamente ao longo do tempo geológico, podendo influenciar a evolução estratigráfica das bacias sedimentares marginais. São apresentados cerca de 60 cenários geodinâmicos mostrando como a variação da estrutura reológica do manto influencia a evolução térmica da litosfera e consequentemente, a história de subsidência da margem. / This work aims to study the dynamics of the upper mantle in continental margins by using numerical models that simulate convective and conductive processes in geological time scale. It was evaluated the contribution of convection and conduction for subsidence history of sedimentary basins. As first tests, simple numerical scenarios with different Rayleigh number were compared with analytic solutions, verifying the validate of the computational solutions. These numerical experiments were followed by the analysis of the lithospheric stability in the geological time scale for different values of viscosity. These experiments were used as a base for the choice of the rheological parameters of the mantle for the models in the context of divergent margins. From the numerical scenarios that better reproduced the lithospheric structure of the Earth, new scenarios were created to simulate the thermal and isostatic evolution of continental margins. As an example, geophysical and geological data extracted from the literature for the sedimentary basin of the Gulf of Lion, Southeastern France, were compared with the results of different geodynamic models published in the literature and with the numerical scenarios obtained in the present work. We observed that the effect of the astenospheric convection preserves the thermal structure of the stretched margin for a long time in comparison with purely conductive models. This implies that, possibly, other processes must be taken into account, such as the effect of the three-dimensional geometry of the Gulf of Lion margin that may contribute to a higher subsidence of the margin than the one obtained in the present work. Additionally, it was observed that mantle convection may induce stress at the base of the lithosphere that dynamically moves it in the geological time, and may influence the stratigraphic evolution of sedimentary basins. It is presented about 60 scenarios showing how the variation of the rheological structure of the mantle is taken into account in the thermal evolution of the lithosphere and consequently in the subsidence history of the margin.

La théorie de la tectonique des plaques permet de décrire les mouvements de premier ordre qui opèrent à la surface de la Terre. S'il est acquis que la convection dans le manteau terrestre en est le moteur, les liens entre les phénomènes profonds et les caractéristiques tectoniques de la surface restent largement méconnus. Jusqu'à très récemment, les modèles de convection du manteau terrestre ne produisaient pas de tectonique de surface pouvant être comparée à celle de la Terre. Récemment, des modèles globaux de convection qui reproduisent une tectonique de surface comparable à la Terre au premier ordre ont été mis au point. Ces modèles produisent des courants mantelliques ascendants et descendants de grande échelle et des déformations localisées en surface dans les zones de divergence et les zones de convergence. Ils génèrent une expansion des fonds océaniques de manière auto-cohérente proche de celle reconstruite pour les 200 derniers millions d'années de l'histoire de la Terre et une dérive de continents similaire à celle observée grâce au paléomagnétisme. Cette thèse s'inscrit parmi les premières tentatives d'utilisation de modèles de convection sphériques auto-organisés à des fins de compréhension de la tectonique de surface. La tectonique produite dans ce type de modèles de convection sera caractérisée finement à travers l'étude des limites de plaques, de leur agencement et de leurs vitesses de déplacement. L'objectif est de pouvoir comparer qualitativement et quantitativement les résultats des calculs de convection avec les reconstructions des mouvements de la surface terrestre grâce à la tectonique des plaques et aux observations de terrain. Dans cette optique, les limites tectoniques ont été définies à la main dans un premier temps afin de comprendre la physique qui gouverne l'agencement caractéristique des plaques tectoniques terrestres. En effet, celle-ci est composée de sept grandes plaques et plusieurs petites dont la répartition statistique indique deux processus de mise en place distincts. Nous avons déterminé les processus responsables de la mise en place de l'agencement caractéristique des plaques tectoniques en surface en faisant varier la résistance de la lithosphère. Plus la lithosphère est résistante, plus la longueur totale et la courbure des zones de subduction diminue à la surface des modèles. Cela s'accompagne également d'une diminution du nombre de petites plaques. En étudiant la fragmentation au niveau des jonctions triples, nous avons montré que les petites plaques étaient associées aux géométries courbées des fosses océaniques. En revanche, les grandes plaques sont contrôlées par les grandes longueurs d'onde de la convection mantellique. Ces deux processus impliquent deux temps de réorganisation, c'est-à-dire l'apparition et la disparition d'une plaque plongeante dans le manteau terrestre (environ 100 millions d'années) pour les grandes plaques, alors que l'échelle de temps de réorganisation des petites plaques dépend des mouvements des fosses et est ainsi plus rapide d'un ordre de grandeur. Afin d'effectuer des analyses quantitatives rapides, des méthodes d'analyse automatique de la surface et de l'intérieur des modèles ont été développées. La première technique concerne la détection automatique des plaques tectoniques à la surface des modèles (ADOPT). ADOPT est un outil de détection basé sur une technique de segmentation d'images utilisée pour détecter des bassins versants. Les champs à la surface des modèles sont transformés en reliefs, soit directement, soit après un processus de filtrage. Cette détection permet d'obtenir des polygones de plaques comparable aux analyses réalisées à la main. Une autre technique de détection a été mise au point pour étudier les panaches mantelliques [etc...] / Plate tectonics theory describes first order surface motions at the surface of the Earth. Although it is agreed upon that convection in the mantle drives the plates, the relationships between deep dynamics and surface tectonics are still largely unknown. Until recently, mantle convection models could not produce surface tectonics that could be compared to that of the Earth. New global models are able to form large-scale ascending and descending mantle currents, as well as narrow regions of localized deformation at the surface where convergence and divergence occur. These models selfconsistently generate an expansion of the oceanic floor similar to that of the last 200 million years on Earth, and continental drift similar to what can be reconstructed with palaeomagnetism. This Ph.D. thesis constitutes one of the first attempts to use self-organised, spherical convection models in order to better understand surface tectonics. Here, the tectonics produced by the models is finely charaterized through the study of plate boundaries, their organisation and their velocities. The goal is to be able to compare qualitatively and quantitatively the results of convection computations with surface motions, as reconstructed using the rules of plate tectonics and field observations. Plate boundaries emerging from the models were first traced and analyzed by hand so as to understand the physics that govern the typical organization of the tectonics plates on Earth. It is characterised by seven large plates and several smaller ones, following a statistical distribution that suggests that two distinct physical processes control the plates’ layout. We have determined the processes responsible for this distribution while varying the strength of the lithosphere (the yield stress). In our models, the stronger the lithosphere, the greater the total subduction length and their curvature, and the fewer the small plates. By studying surface fragmentation with triple junctions, we showed that the formation of small plates is associated with oceanic trench curvature. Large plates, however, are controlled by the long wavelengths of the convection cells. These two processes involve two different reorganisation times, controlled either by the accretion and the subduction of the large plates (about 100 Myrs), or by trench motions for the smaller plates. In order to improve the efficiency of our analysis, we have developed automated methods to study the surface and the interior of the models. The first technique is about detecting the tectonic plates automatically at the surface of the models. It is called ADOPT. It is a tool based on image segmentation technique to detect the watersheds. The surface fields of the convection models are converted into a relief field, either directly or using a distance method. This automatic detection allows to obtain plates polygons similar to the hand analysis. Another technique of detection has been developed to study mantle plumes. These analyzes were used to determine the driving forces behind the plates layout, to quantify the timing of reorganizations and to evaluate the implication of the models rheology on the surface distribution. These new analytical tools and the constant evolution of the quality of mantle convection models allow us to improve our understanding of the link between mantle dynamics and surface tectonics, but also to target necessary improvements in the convection models used

Many highly developed physical models poorly approximate actual physical systems due to natural random noise. For example, convection in the earth's mantle—a fundamental process for understanding the geochemical makeup of the earth's crust and the geologic history of the earth—exhibits chaotic behavior, so it is difficult to model accurately. In addition, it is impossible to directly measure temperature and fluid viscosity in the mantle, and any indirect measurements are not guaranteed to be highly accurate. Over the last 50 years, mathematicians have developed a rigorous framework for reconciling noisy observations with reasonable physical models, a technique called data assimilation. We apply data assimilation to the problem of mantle convection with the infinite-Prandtl Boussinesq approximation to the Navier-Stokes equations as the model, providing rigorous conditions that guarantee synchronization between the observational system and the model. We validate these rigorous results through numerical simulations powered by a flexible new Python package, Dedalus. This methodology, including the simulation and post-processing code, may be generalized to many other systems. The numerical simulations show that the rigorous synchronization conditions are not sharp; that is, synchronization may occur even when the conditions are not met. These simulations also cast some light on the true relationships between the system parameters that are required in order to achieve synchronization. To conclude, we conduct experiments for two closely related data assimilation problems to further demonstrate the limitations of the rigorous results and to test the flexibility of data assimilation for mantle-like systems.

The thermal histories of terrestrial planets are investigated using two parameterized mantle convection models for either Earth like planets and planets with no active plate tectonics. Using parameterized models of mantle convection, we performed computer simulations of planetary cooling and volatile cycling. The models estimate the amount of volatile in mantle reservoir, and calculate the outgassing and regassing rates. A linear model of volatile concentration-dependent is assumed for the activation energy of the solid-state creep in the mantle. The kinematic viscosity of the mantle is thus dynamically affected by the activation energy through a variable concentration in volatile. Mantle temperature and heat flux is calculated using a model derived from classic thermal boundary layer theory of a single layered mantle with temperature dependent viscosity. The rate of volatile exchanged between mantle and surface is calculated by balancing the amount of volatiles degassed in the atmosphere by volcanic and spreading related processes and the amount of volatiles recycled back in the mantle by the subduction process. In the cases that lack plate tectonics, the degassing efficiency is dramatically reduced and the regassing process is absent. The degassing effect is dependent on average spreading rate of tectonic plates and on the amount of volatile in the melt extract in the transition zone between mantle and upper boundary laver. The regassing effect is dependent on the subduction rate and on the amount of volatile present on a hydrated layer on top of the subducting slab. The degassing and regassing parameters are all related to the intensity of the convection in the mantle and to the surface temperature of the planet, and they are regulated by the amount of volatiles in reservoir. Comparative study with the previous models display significant differences and improve the versatility of the model. The optimum efficiency factors found are in the range of 0.01-0.06 for degassing/regassing processes, in agreement with more recent estimates. An important effect of the volatile cycling process is a general negative feedback effect that results in a general trend to adjust the mantle volatile content in time to a value set by the energy balance in the system. As a result, the initial amount of volatile in the mantle is rendered irrelevant for late stage of thermal evolution. In the case of no plate tectonics, the opposite effect takes place: initial volatilization plays an important role through entire evolution. The implications of mantle convection on the stability of the lithosphere were investigated further using the thermal history calculations and numeric simulations. They point to the conclusion that mantle convection induced stress levels increase from the past to the present fact that leads to a greater potential of craton deformation. The main consequence of this trend is that sections of continental lithosphere that have remained stable since the Archean and Proterozoic are becoming progressively more prone to instability in the geologically modem era. After the volatiles are degassed from the mantle, they are cycled through the atmosphere. The interact with the climate influencing the surface temperature, and further controlling the mantle convection. Using a grey radiative-convective model for the atmosphere, we analyzed the feedback relationships between volatiles, especially water, and surface temperature. We showed that large amount of water degassed during a hot, possible melt ocean phase after the planet formation could conserve large amount of water in atmosphere and maintain the surface temperature at moderate level.