Thermal histories of the core and mantle

Recognition that the cooling of the core is accomplished by conduction of heat into a thermal boundary layer (D″) at the base of the mantle, partly decouples calculations of the thermal histories of the core and mantle. Both are controlled by the temperature-dependent rheology of the mantle, but in different ways. Thermal parameters of the Earth are more tightly constrained than hitherto by demanding that they satisfy both core and mantle histories. We require evolution from an early state, in which the temperatures of the top of the core and the base of the mantle were both very close to the mantle solidus, to the present state in which a temperature increment, estimated to be ～ 800 K, has developed across D″. The thermal history is not very dependent upon the assumption of Newtonian or non-Newtonian mantle rheology. The thermal boundary layer at the base of the mantle (i.e., D″) developed within the first few hundred million years and the temperature increment across it is still increasing slowly. In our preferred model the present temperature at the top of the core is 3800 K and the mantle temperature, extrapolated to the core boundary without the thermal boundary layer, is 3000 K. The mantle solidus is 3860 K. These temperatures could be varied within quite wide limits without seriously affecting our conclusions. Core gravitational energy release is found to have been remarkably constant at ～ 3 × 10¹¹ W. nearly 20% of the core heat flux, for the past 3 × 10⁹ y, although the total terrestrial heat flux has decreased by a factor of 2 or 3 in that time. This gravitational energy can power the “chemical” dynamo in spite of a core heat flux that is less than that required by conduction down an adiabatic gradient in the outer core; part of the gravitational energy is used to redistribute the excess heat back into the core, leaving 1.8 × 10¹¹ W to drive the dynamo. At no time was the dynamo thermally driven and the present radioactive heating in the core is negligibly small. The dynamo can persist indefinitely into the future; available power 10¹⁰ y from now is estimated to be 0.3 × 10¹¹ W if linear mantle rheology is assumed or more if mantle rheology is non-linear. The assumption that the gravitational constant decreases with time imposes an implausible rate of decrease in dynamo energy. With conventional thermodynamics it also requires radiogenic heating of the mantle considerably in excess of the likely content of radioactive elements.     

The cooling Earth: A reappraisal

By treating the lithosphere as a diffusive boundary layer to mantle convection, the convective speed or mantle creep rate, $\text{?}\text{?}$, can be related to the mantle-derived heat flux, $\text{Q}\text{?}$. If cell size is independent of $\text{Q}\text{?}{}^2$ then $\text{?}\text{?}\text{∝}\text{Q}\text{?}$. (If cell size varies with $\text{Q}\text{?}$, then a different power law prevails, but the essential conclusions are unaffected.) Then the factthat for constant thermodynamic efficiency of mantle convection, the mechanical power dissipation is proportionalto $\text{Q}\text{?}$, gives convective stress $\text{σ ∝}\text{Q}\text{?}{}^{\mathrm{?1}}$, i.e. the stress increases as the convection slows. This means an increasing viscosityor stiffness of the mantle which can be identified with a cooling rate in terms of a temperature-dependent creep law. If we suppose that the mantle was at or close to its melting point within 1 or 2 × 10⁸ years of accretionof the Earth, the whole scale of cooling is fixed. The present rate of cooling is estimated to be about 4.6 × 10^?8 deg y^?1 for the average mantle temperature, assumed to be 2500 K, but this very slow cooling rate represents a loss ofresidual mantle heat of 7 × 10¹² W, about 30% of the total mantle-derived heat flux. This conclusion requires theEarth to be deficient in radioactive heat, relative to carbonaceous chondrites. A consideration of mantle outgassing and atmospheric argon leads to the conclusion that the deficiency is due to depletion of potassium, and that the K/U ratio of the mantle is only about 2500, much less than either the crustal or carbonaceous chondritic values. Thetotal terrestrial potassium is estimated to be about 6 × 10²⁰ kg. Acceptance of the cooling of the Earth removes the necessity for potassium in the core.     

Convection and lunar thermal history

The effect of solid convection on the thermal evolution of the Moon is explored for a variety of viscosities, radioactive differentiation efficiencies and initial temperature profiles. Convective heat flux in the models is calculated using an empirical relation derived from the results of laboratory experiments and numerical solutions of the Navier-Stokes equations. The method retains the spherically symmetric approximation and, therefore, greatly facilitates numerical calculations.Results show that even though solid convection may determine the thermal state of the lunar interior, it does not necessarily produce a quasi-steady thermal balance between heat sources and surface loss. An imbalance persists, due to the cooling and growth of the nonconvecting lithosphere. The state of the lithosphere is sensitive to the efficiency of heat source redistribution, while that of the convecting interior depends primarily on rheology. Convecting models have viscosities of 10²¹–10²² cm²s^?1 in their interiors; the central temperature must be above 1100°C. Convection occurring within the first billion years after formation could have led to mare flooding by magma produced in hot zones of convection cells. However, it cannot be shown from model calculations alone that solid convection must have dominated lunar thermal history.     

Comparison of Early Evolutions of Mimas and Enceladus

Thermal history of Mimas and Enceladus is investigated from the beginning of accretion to 400 Myr. The numerical model of convection combined with the parameterized theory is used. The following heat sources are included: short lived and long lived radioactive isotopes, accretion, serpentinization, and phase changes. The heat transfer processes are: conduction, solid state convection, and liquid state convection. We find that temperature of Mimas’ interior was significantly lower than that of Enceladus. If Mimas accreted 1.8 Myr after CAI then the internal melting and differentiation did not occur at all. Comparison of thermal models of Mimas and Enceladus indicates that conditions favorable for the start of tidal heating lasted for a short time (~10⁷ yr) in Mimas and for ~10⁸ yr in Enceladus. This could explain the Mimas—Enceladus paradox. In fact, in view of the chronology based on cometary impact rate, one cannot discard a possibility that also Mimas was for some time active and it has the interior differentiated on porous core and icy mantle.     

The structure of convection in the spherical mantle with internal heating

Thermal convection in the mantle is caused by the heat transported upwards from the core and by the heat produced by the internal radioactive sources. According to the data on the heat transfer by the mantle plumes and geochemical evidence, only 20% of the total heat of the Earth is supplied to the mantle from the core, whereas most of the heat is generated by the internal sources. Along with the models that correctly allow for the internal heat sources, there are also many publications (including monographs) on the models of mantle convection that completely ignore the internal heating or the heat flux from below. In this study, we analyze to what extent these approximations could be correct. The analytical distributions of temperature and heat flux in the case of internal heating without convection and the results of the numerical modeling for convection with different intensity are presented. It is shown that the structure of thermal convection is governed by the distribution of the heat flux in the mantle but not by the heat balance, as it is typically implicitly assumed in most works. Heat production by the internal sources causes the growth of the heat flux as a function of radius. However, in the spherical mantle of the Earth, the heat flux decreases with radius due to the geometry. It turned out that with the parameters of the present Earth, both these effects compensate each other to a considerable extent, and the resulting heat flux turns out to be nearly constant as a function of radius. Since the structure of the convective flows in the mantle is determined by the distributions of heat flux and total heat flux, in the Cartesian models of the mantle convection the effective contribution of internal heating is small, and ignoring the heat flux from the core significantly distorts the structure of the convective currents and temperature distributions in the mantle.     

Heat transport by variable viscosity convection and implications for the Earth's thermal evolution

The case is presented that the efficiency of variable viscosity convection in the Earth's mantle to remove heat may depend only very weakly on the internal viscosity or temperature. An extensive numerical study of the heat transport by 2-D steady state convection with free boundaries and temperature dependent viscosity was carried out. The range of Rayleigh numbers (Ra) is 10⁴?10⁷ and the viscosity contrast goes up to 250000. Although an absolute or relative maximum of the Nusselt number (Nu) is obtained at long wavelength in a certain parameter range, at sufficiently high Rayleigh number optimal heat transport is achieved by an aspect ratio close to or below one. The results for convection in a square box are presented in several ways. With the viscosity ratio fixed and the Rayleigh number defined with the viscosity at the mean of top and bottom temperature the increase of Nu with Ra is characterized by a logarithmic gradient β = ?ln(Nu)/? ln(Ra) in the range of 0.23–0.36, similar to constant viscosity convection. More appropriate for a cooling planetary body is a parameterization where the Rayleigh number is defined with the viscosity at the actual average temperature and the surface viscosity is fixed rather than the viscosity ratio. Now the logarithmic gradient β falls below 0.10 when the viscosity ratio exceeds 250, and the velocity of the surface layer becomes almost independent of Ra. In an end-member model for the Earth's thermal evolution it is assumed that the Nusselt number becomes virtually constant at high Rayleigh number. In the context of whole mantle convection this would imply that the present thermal state is still affected by the initial temperature, that only 25–50% of the present-day heat loss is balanced by radiogenic heat production, and the plate velocities were about the same during most of the Earth's history.     

Thermal regime of the Earth's core

The outer core is assumed to consist of iron and sulfur, with a small amount of potassium that generates heat by radioactive decay of sim||pre|40 K. Two cases are considered, corresponding respectively to a high rate of heat production (Q = 2 · 10¹² cal./sec, about 0.1% K), and to a low rate (Q = 2 · 10¹¹ cal./sec). The temperature at a depth of 2800 km in the mantle is taken to be 3300°K (Wang, 1972). The temperature T_c at the core-mantle boundary depends on whether or not a density gradient in the lowermost layer D″ of the mantle prevents convection in that layer. In the first case, and for high Q, T_c = 4500–5000°K. In the second case, or for low Q, T_c ≈ 3500°K.The heat-conduction equation is used to calculate the temperature T_i at the inner-core boundary in the absence of convection. For high Q, T_i ? T_c ≈ 1600°K; for low Q, T_i ? T_c ≈ 160°K. Corresponding temperature gradients at r = r_c and r = r_i are listed in Table I.The adiabatic gradient at the top of the core is calculated by the method of Stewart (1970). It strongly depends on the parameters (ρ₀, c₀, γ₀, etc.) that characterize core material at low pressure. Stewart has drawn graphs that allow the selection of sets of parameters that are consistent with seismic velocities and a given density distribution in the core. Some acceptable sets of parameters are listed in Table II. Many sets yield temperatures T_c in the range 3500–5000°K; some give an adiabatic gradient steeper than the conductive gradient and are compatible with convection; others do not. Since properties of FeS melts remain unknown, there is at present no way of selecting any set in preference to another.Properties of the FeS system at low pressure suggest the possible appearance of immiscibility at high temperature in liquids of low sulfur content; accordingly, the inner-core boundary is thought to represent equilibrium between a solid (FeNi) inner core and a liquid layer containing only a small amount of sulfur; layer F in turn is in equilibrium with another liquid (forming layer E) containing more sulfur, and slightly less dense, than F. The temperature T_i at the inner-core boundary is about 6000–6500°K for high Q and T_c ≈ 4500–5000°K. It is consistent with Alder's (1966) and Leppaluoto's (1972) estimates of the melting point of iron at 3.3 Mbar, but not with that of Higgins and Kennedy (1971).     

A heat flow survey on anomaly M0 south of the Bermuda Rise

We have obtained a suite of 42 closely spaced, acoustically navigated, heat flow measurements on well-sedimented crust of anomaly M0 age (109 Ma) in the northwest Atlantic Ocean (25°N, 68°W; 950 km south of Bermuda). The mean and standard deviation of the values obtained are 1.13 HFU (μcal/cm² s) (47.3 mW/m²) and 0.05 HFU (2.1 mW/m²), respectively. Some of the variability is accounted for by refractive effects of the basement topography. Drill core data and our modelling suggest that the thermal conductivity contrast between sediments and basement rocks in this region is less than a factor of 1.6. The mean heat flow is close to the 1.1 HFU (46 mW/m²) predicted by both the plate and boundary layer cooling models of the oceanic lithosphere. This is the first detailed comparison with theoretical cooling models on old Atlantic Ocean crust. Since the difference in surface heat flow (0.15 HFU) predicted by the two cooling models for the oldest observed oceanic lithosphere (180 Ma) is also not much larger than the range of uncertainty in our observations, discrimination between the two models on the basis of surface heat flow data alone may prove difficult.     

Temperature production in the continental crust due to radioactive heat sources

Summary The purpose of this paper is to present a method for computing the temperature distribution in the continental crust as a function of depth and time. Computed results indicate that the rise of the temperature due to the decay of long half-life radioactive elements K⁴⁰, U²³⁵, U²³⁸, and Th²³² will reach its maximum 90 m.y. after heat sources and temperature are redistributed in the upper part of the earth. This suggests that the lower part of the crust and upper mantle were periodically remelted before the present stage of the crust was reached. At the present time the decay of the radioactive elements U²³⁸ and Th²³² still significantly influences the crustal temperature. The present near-surface thermal gradient due to the radioactive heat sources alone may reach 20°C/km in a high radioactive heat source province.     

Subsurface temperature measurements and terrestrial heat flow estimates in the Aswan region,Egypt

Summary The results of temperature measurements in six boreholes, drilled near the west shores of the Lake Nasser in the Aswan region in the Nubian Desert, are reported. Temperature-depth profiles, complemented by thermal conductivity determinations of surface rock samples, were used to calculate the existing range of heat flow density in the region as 40 ÷ 60 mWm⁻², which is consistent with the values from the adjacent areas, reported in the literature. The extrapolated mean annual ground surface temperatures amount, on the average, to 30°C, which is 3 K higher than the mean annual air temperature in the region.     

Numerical analysis for cooling effect of open boundary ripped-rock embankment on Qinghai-Tibetan railway

At present, the Qinghai-Tibetan railway is being built, and it will pass across more than 550-km perma-frost regions. Therefore, the key to the stability of therailway embankment lies in solving the permafrost problem. Because global warming and existence of railway tend to degrade the permafrost in these re-gions[1], more difficulties and problems are induced in the construction and maintenance of railway. In the area where the mean annual air temperature is higher than a certain value, the …     

An anomalous cooling event observed in the Bay of Bengal during June 2009

Sea surface temperature (SST) variability over the Bay of Bengal (BoB) has the potential to trigger deep moist convection thereby affecting the active-break cycle of the monsoons. Normally, during the summer monsoon season, SST over the BoB is observed to be greater than 28°C which is a pre-requisite for convection. During June 2009, satellite observations revealed an anomalous basin-wide cooling and the month is noted for reduced rainfall over the Indian subcontinent. In this study, we analyze the likely mechanisms of this cooling event using both satellite and moored buoy observations. Observations showed deepened mixed layer, stronger surface currents, and enhanced heat loss at the surface in the BoB. Mixed layer heat balance analysis is carried out to resolve the relative importance of various processes involved. We show that the cooling event is primarily induced by the heat losses at the surface resulting from the strong wind anomalies, and advection and vertical entrainment playing secondary roles.     

鄂尔多斯盆地深部热结构特征及其对华北克拉通破坏的启示

基于二维稳态热传导方程,利用有限元数值模拟方法,选取东西向横穿鄂尔多斯盆地地质与地球物理解释大剖面进行了深部温度场数值模拟研究,得到了华北克拉通西部的鄂尔多斯盆地下伏岩石圈热结构特征.地幔热流变化范围:21.2~24.5mW·m-2,体现为东高西低特征.壳幔热流比(Qc/Qm)介于1.51~1.84之间,为"热壳冷幔".与华北东部地幔热流对比表明,西部的鄂尔多斯盆地相对处于稳定的深部动力学环境.在岩石圈热结构研究基础上,对克拉通地震岩石圈与热岩石圈厚度差异进行了对比,研究表明:鄂尔多斯盆地西部地震岩石圈与热岩石圈厚度差异约达140km,而东部的汾渭地堑,渤海湾盆地二者差异逐渐减小.华北克拉通自西向东,地震岩石圈厚度与热岩石圈厚度差异不断减小,意味着华北克拉通岩石圈下部的软流圈地幔黏性系数自西向东逐渐降低,本文从地热学角度可能印证了太平洋俯冲脱水作用对华北克拉通的影响.     

青藏高原东南缘热流估算及与地震活动相关性分析

青藏高原东南缘地区内部构造运动强烈,是地热资源发育与地震事件频发的活动地区.大地热流记录了发生在地球深部各种作用过程的热学信息,可以作为地质构造活动和地震活动研究的有效约束,但是大范围的热流数据测量很难实现,因此,本文根据居里面深度结合放射性元素分布等计算了青藏高原东南缘的大地热流分布.首先,通过地表放射性元素的分布计算出地表产热量的分布,然后,利用相关地热参数之间的关系迭代计算出该地区地壳上下层的热导率分布,最终估算出地表热流及地下不同深度处热流值的分布.本文结果表明:(1)青藏高原东南缘的大地热流位于44～108mW·m~(-2)之间,平均75mW·m~(-2),符合研究地区西南高、东北低的背景趋势,地壳内部热流值随深度的增加而降低.大部分地区地表热流异常与实际地热带分布相吻合,如川西、藏东南与滇西地区等地为地热高值区,川东和楚雄等地为热流低值区.(2)结合其他地球物理探测结果,总结了地壳内部热流与地震事件的联系:在地热梯度带地区,当两侧地层在一定深度范围内存在明显物性差异时,地震事件高发.     

Continental growth and volatile exchange during Earth''s evolution

We investigate the thermal and degassing history of the Earth with the help of a parameterized mantle convection model including the volatile exchange between mantle and surface reservoirs. The weakening of mantle silicates by dissolved volatiles is described by a functional relationship between creep rate and water fugacity. We use flow law parameters of diffusion creep in olivine under dry and wet conditions. The mantle degassing rate is considered as directly proportional to the seafloor spreading rate, which is also dependent on the mantle heat flow and the continental area. To calculate the spreading rate, we assume three different continental growth models: constant growth, delayed growth, and the one proposed by Reymer and Schubert (1984, Tectonics, 3: 63–77). The rate of regassing also depends on the seafloor spreading rate, as well as on other factors. Both mechanisms (degassing and regassing) are coupled self-consistently with the help of a parameterized convection model under implementation of a temperature and volatile-content dependent mantle viscosity. We calculate time series for the Earth's evolution over 4.6 Gyr for the average mantle temperature, the mantle heat flow, the mantle viscosity, the Rayleigh number, the Urey ratio, the volatile loss, and the seafloor spreading rate. In those numerical simulations with continental growth from the beginning and a high initial average mantle temperature water is outgassed rapidly. In the delayed continental growth model there is a very early outgassing event and the delayed continental growth has no remarkable influence on the thermal and outgassing history. A similar situation is found for the linear continental growth model but not for the Reymer and Schubert (1984) model.     

Core cooling by subsolidus mantle convection

Although vigorous mantle convection early in the thermal history of the Earth is shown to be capable of removing several times the latent heat content of the core, we are able to construct a thermal evolution model of the Earth in which the core does not solidify. The large amount of energy removed from the model Earth's core by mantle convection is supplied by the internal energy of the core which is assumed to cool from an initial high temperature given by the silicate melting temperature at the core-mantle boundary. For the smaller terrestrial planets, the iron and silicate melting temperatures at the core-mantle boundaries are more comparable than for the Earth, and the cores of these planets may not possess enough internal energy to prevent core solidification by mantle convection. Our models incorporate temperature-dependent mantle viscosity and radiogenic heat sources in the mantle. The Earth models are constrained by the present surface heat flux and mantle viscosity. Internal heat sources produce only about 55% of the Earth model's present surface heat flow.     

Thickness and thermal history of continental crust and root zones

The survival to the present of the Archean nuclei of Precambrian shields requires special explanation if, as seems likely, the rate of heat flow out of the earth was two or three times greater in the late Archean (2.5 b.y. ago) than at present, since such a high heat flux would have melted the base of the Archean crust. It is proposed that there must have existed beneath stable continental crust a root zone (or lithosphere, or tectosphere) at least 200 km thick which has acted as a thermal buffer between the crust and the convecting mantle; this is virtually the same model as has been proposed to explain the present distribution of heat flow between continents and oceans. The strong temperature dependence of silicate rheology insures that the mantle temperature at the base of the root zone was no more than about 150°C higher in the late Archean than at present; the greater Archean heat flux would have been removed mainly through faster sea-floor spreading. To have survived, the root zone must be mechanically and chemically distinct from the rest of the mantle, and its formation was probably intimately related to the differentiation and stabilization of the continental crust.     

Constraints on the dynamics of mantle plumes from uplift of the Hawaiian Islands

The ∼0.2 mm/yr uplift of Hawaiian islands Lanai and Molokai and Hawaiian swell topography pose important constraints on the structure and dynamics of mantle plumes. We have formulated 3-D models of mantle convection to investigate the effects of plume-plate interactions on surface vertical motions and swell topography. In our models, the controlling parameters are plume radius, excess plume temperature, and upper mantle viscosity. We have found that swell height and swell width constraints limit the radius of the Hawaiian plume to be smaller than 70 km. The additional constraint from the uplift at Lanai requires excess plume temperature to be greater than 400 K. If excess plume temperature is 400 K, models with plume radius between 50 and 70 km and upper mantle viscosity between 10²⁰ and 3×10²⁰ Pa s satisfy all the constraints. Our results indicate that mantle plume in the upper mantle may be significantly hotter than previously suggested. This has important implications for mantle convection and mantle melting. In addition to constraining plume dynamics, our models also provide a mechanism to produce the observed uplift at Lanai and Molokai that has never been satisfactorily explained before.     

Estimate of influence of U-Th-K radiogenic heat on cooling process of granitic melt and its geological implications

The U-Th-~(40)K concentrations of granite are on 1―2 orders of magnitude greater than those of basal- tic-ultrabasic rocks. Radiogenic heat of a granitic melt has significant influence on the cool- ing-crystallization period of the melt. In this paper we derived a formula to calculate prolongation period (tA) of cooling-crystallization of a granitic melt caused by radiogenic heat. Calculation using this for- mula and radioactive element concentrations (U=5.31×10-6; Th=23.1×10-6; K=4.55%) for the biotite adamellite of the Jinjiling batholith shows that the tA of the adamellite is 1.4 times of the cooling period of the granitic melt without considering radiogenic heat from the initial temperature (Tm=960℃) to crystallization temperature (Tc=600℃) of the melt. It has been demonstrated that the radiogenic heat produced in a granitic melt is a key factor influencing the cooling-crystallization process of the granitic melt, and is likely one of the reasons for inconsistence between emplacement ages and crystallization ages of many Meso-Cenozoic granitoids.     

A one-dimensional modeling study of the diurnal cycle in the equatorial Atlantic at the PIRATA buoys during the EGEE-3 campaign

A one-dimensional model is used to analyze, at the local scale, the response of the equatorial Atlantic Ocean under different meteorological conditions. The study was performed at the location of three moored buoys of the Pilot Research Moored Array in the Tropical Atlantic located at 10° W, 0° N; 10° W, 6° S; and 10° W, 10° S. During the EGEE-3 (Etude de la circulation océanique et de sa variabilité dans le Golfe de Guinee) campaign of May–June 2006, each buoy was visited for maintenance during 2 days. On board the ship, high-resolution atmospheric parameters were collected, as were profiles of temperature, salinity, and current. These data are used here to initialize, force, and validate a one-dimensional model in order to study the diurnal oceanic mixed-layer variability. It is shown that the diurnal variability of the sea surface temperatures is mainly driven by the solar heat flux. The diurnal response of the near-surface temperatures to daytime heating and nighttime cooling has an amplitude of a few tenths of degree. The computed diurnal heat budget experiences a net warming tendency of 31 and 27 W m⁻² at 0° N and 10° S, respectively, and a cooling tendency of 122 W m⁻² at 6° S. Both observed and simulated mixed-layer depths experience a jump between the nighttime convection phase and the well-stabilized diurnal water column. Its amplitude changes dramatically depending on the meteorological conditions occurring at the stations and reaches its maximum amplitude (~50 m) at 10° S. At 6° and 10° S, the presence of barrier layers is observed, a feature that is clearer at 10° S. Simulated turbulent kinetic energy (TKE) dissipation rates, compared to independent microstructure measurements, show that the model tracks their diurnal evolution reasonably well. It is also shown that the shear and buoyancy productions and the vertical diffusion of TKE all contribute to the supply of TKE, but the buoyancy production is the main source of TKE during the period of the simulation.