The Boussinesq and anelastic liquid approximations for convection in the Earth’s core

Convection in the Earth’s core is usually studied in the Boussinesq approximation in which the compressibility of the liquid is ignored. The density of the Earth’s core varies from ICB to CMB by approximately 20%. The question of whether we need to take this variation into account in core convection and dynamo models is examined. We show that it is in the thermodynamic equations that differences between compressible and Boussinesq models become most apparent. The heat flux conducted down the adiabat is much smaller near the inner core boundary than it is near the core-mantle boundary. In consequence, the heat flux carried by convection is much larger nearer the inner core boundary than it is near the core-mantle boundary. This effect will have an important influence on dynamo models. Boussinesq models also assume implicitly that the rate of working of the gravitational and buoyancy forces, as well as the Ohmic and viscous dissipation, are small compared to the heat flux through the core. These terms are not negligible in the Earth’s core heat budget, and neglecting them makes it difficult to get a thermodynamically consistent picture of core convection. We show that the usual anelastic equations simplify considerably if the anelastic liquid approximation, valid if αT?1, where α is the coefficient of expansion and T a typical core temperature, is used. The resulting set of equations are not significantly more difficult to solve numerically than the usual Boussinesq equations. The relationship of our anelastic liquid equations to the Boussinesq equations is also examined.     

Entrainment from a bed of particles by thermal convection

Differentiation in magma chambers, in the Earth's core and in the partially molten early Earth is a competitive process between sedimentation and re-entrainment of crystals in the presence of convection. Previous studies show that the particles suspended in convective layers eventually settle and do so almost as fast as in the absence of convection. However, the nature and magnitude of the competing entrainment has remained unclear. Here we provide a simple theory and experimental evidence showing that entrainment occurs at the crests of dunes created in the particle bed at the base of a convecting fluid. In both laminar and turbulent regimes, the dune formation and entrainment are driven by viscous stresses produced by thermal plumes. At sufficiently high Rayleigh numbers the particles are probably entrained by Reynolds stresses. Entrainment in the Earth's core is hardly possible because it requires unreasonably small crystals. Entrainment of 10⁻²–10⁻¹ cm diameter crystals is very likely in magma oceans. For magma chambers entrainment requires large viscosities (> 10⁶ P) and even when it occurs, the total amount of the suspended solid fraction is very small.     

The rapid dissipation of magnetic fields in highly conducting fluids

Abstract

This paper treats the dynamical conditions that obtain when long straight parallel twisted flux tubes in a highly conducting fluid are packed together in a broad array. It is shown that there is generally no hydrostatic equilibrium. In place of equilibrium there is a dynamical nonequilibrium, leading to neutral point reconnection and progressive coalescence of neighboring tubes (with the same sense of twisting), forming tubes of larger diameter and reduced twist. The magnetic energy in the twisting of each tube declines toward zero, dissipated into small-scale motions of the fluid and thence into heat.

The physical implications are numerous. For instance, it has been suggested that the subsurface magnetic field of the sun is composed of close-packed twisted flux tubes. Any such structures are short lived, at best.

The footpoints of the filamentary magnetic fields above bipolar magnetic regions on the sun are continually shuffled and rotated by the convection, so that the fields are composed of twisted rubes. The twisting and mutual wrapping is converted directly into fluid motion and heat by the dynamical nonequilibrium, so that the work done by the convection of the footpoints goes directly into heating the corona above. This theoretical result is the final step, then, in understanding the assertion by Rosner, Tucker, and Valana, and others, that the observed structure of the visible corona implies that it is heated principally by direct dissipation of the supporting magnetic field. It is the dynamical nonequilibrium that causes the dissipation, in spite of the high electrical conductivity. It would appear that any bipolar magnetic field extending upward from a dense convective layer into a tenuous atmosphere automatically produces heating, and a corona of some sort, in the sun or any other convective star.     

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.     

Self‐convection of floating heat sources: A model for continental drift

Abstract

Two models of floating heat sources are studied. In the first model the motion of two line heat sources constrained to float at an arbitrary depth in a viscous fluid is determined in the limit of small convection velocities. It is found that the sources drift apart and at great separation attain a constant velocity proportional to the square root of the heat flux. The second model is a floating block heat source, presumed to be very long compared to its depth. It is found to exhibit periodic excursions between the end walls of the fluid container with the same dependence of velocity on heat flux as the line sources. A series of experiments are described which exhibit various features of the theory. The numerical values found when the theory is applied to the earth suggest that the idealized flows may be useful in the interpretation of continental drift.     

Comments on some aspects of particulate transport in the oceans

Recent studies of dissolved and particulate concentrations of trace elements and radionuclides amply demonstrate the importance of particulate transport in the case of several elements. A significant in-situ addition (J-flux) or removal (J-efflux) occurs in the case of a number of elements. However, to date it is not clear how the particulate processes occur and how the particles themselves are transported. Some of the problems are outlined briefly.It is shown that whereas a substantial flux in the case of some elements is due to transport by consolidated fecal particles, this transport does not generally lead to any substantial in-situ addition to deep waters. Changes in the dissolved concentrations of elements within the oceans occur due to small particles (1–10 μm) which sink stochastically with a mean speed of ～10^?3 cm/s. The larger particles sinking at higher Stokes' velocities impact and carry along the small particles. The smaller particles, ～1 μm size are thus transported down rapidly by the larger particles by apiggy-back mechanism. Simple theoretical calculations are consistent with the measured vertical transport rates based on studies of radionuclides.     

On the current-voltage relationship in fluid theory

The kinetic theory of precipitating electrons with Maxwellian source plasma yields the well-known current-voltage relationship (CV-relationship; Knight formula), which can in most cases be accurately approximated by a reduced linear formula. Our question is whether it is possible to obtain this CV-relationship from fluid theory, and if so, to what extent it is physically equivalent with the more accurate kinetic counterpart. An answer to this question is necessary before trying to understand how one could combine time-dependent and transient phenomena such as Alfvnic waves with a slowly evolving background described by the CV-relationship. We first compute the fluid quantity profiles (density, pressure etc.) along a flux tube based on kinetic theory solution. A parallel potential drop accumulates plasma (and pressure) below it, which explains why the current is linearly proportional to the potential drop in the kinetic theory even though the velocity of the accelerated particles is only proportional to the square root of the accelerating voltage. Electron fluid theory reveals that the kinetic theory results can be reproduced, except for different numerical constants, if and only if the polytropic index γ is equal to three, corresponding to one-dimensional motion. The convective derivative term v∇v provides the equivalent of the “mirror force” and is therefore important to include in a fluid theory trying to describe a CV-relationship. In one-fluid equations the parallel electric field, at least in its functional form, emerges self-consistently. We find that the electron density enhancement below the potential drop disappears because the magnetospheric ions would be unable to neutralize it, and a square root CV-relationship results, in disagreement with kinetic theory and observations. Also, the potential drop concentrates just above the ionosphere, which is at odds with observations as well. To resolve this puzzle, we show that considering outflowing ionospheric ions restores the possibility of having the acceleration region well above the ionosphere, and thus the electron kinetic (and fluid, if γ=3) theory results are reproduced in a self-consistent manner. Thus the inclusion of ionospheric ions is crucial for a feasible CV-relationship in fluid theory. Constructing a quantitative fluid model (possibly one-fluid) which reproduces this property would be an interesting task for a future study.     

Large aspect ratio cells in two-dimensional thermal convection

Numerical experiments have been carried out on two-dimensional thermal convection, in a Boussinesq fluid with infinite Prandtl number, at high Rayleigh numbers. With stress free boundary conditions and fixed heat flux on upper and lower boundaries, convection cells develop with aspect ratios (width/depth) λ? 5, if heat is supplied either entirely from within or entirely from below the fluid layer. The preferred aspect ratio is affected by the lateral boundary conditions. If the temperature, rather than the heat flux, is fixed on the upper boundary the cells haveλ ≈ 1. At Rayleigh numbers of 2.4 × 10⁵ and greater, small sinking sheets are superimposed on the large aspect ratio cells, though they do not disrupt the circulation. Similar two-scale flows have been proposed for convection in the earth's mantle. The existence of two scales of flow in two-dimensional numerical experiments when the viscosity is constant will allow a variety of geophysically important effects to be investigated.     

Mantle-driven geodynamo features – Effects of compositional and narrow D″ anomalies

Lower mantle heterogeneity could cause deviations from axial symmetry in geodynamo properties. Global tomography models are commonly used to infer the pattern of core–mantle boundary heat flux via a linear relation that corresponds to a purely thermal interpretation of lower mantle seismic anomalies, ignoring both non-thermal origins and non-resolved small scales. Here we study the possible impact on the geodynamo of narrow thermal anomalies in the base of the mantle, originating from either compositional heterogeneity or sharp margins of large-scale features. A heat flux boundary condition composed of a large-scale pattern and narrow ridges separating the large-scale positive and negative features is imposed on numerical dynamos. We find that hot ridges located to the west of a positive large-scale core–mantle boundary heat flux anomaly produce a time-average narrow elongated upwelling, a flow barrier at the top of the core and intensified low-latitudes magnetic flux patches. When the ridge is located to the east of a positive core–mantle boundary heat flux anomaly, the associated upwelling is weaker and the homogeneous dynamo westward drift leaks, precluding persistent intense low-latitudes magnetic flux patches. These signatures of the core–mantle boundary heat flux ridge are evident in the north–south component of the thermal wind balance. Based on the pattern of lower mantle seismic tomography (Masters et al., 2000), we hypothesize that hot narrow thermal ridges below central Asia and the Indian Ocean and below the American Pacific coast produce time-average fluid upwelling and a barrier for azimuthal flow at the top of the core. East of these ridges, below east Asia and Oceania and below the Americas, time-average intense geomagnetic flux patches are expected.     

Structure of the inner core boundary

Abstract

A model of the inner-core boundary (ICB) is constructed which is consistent with current ideas of the dynamic and thermodynamic state of the core and which is capable of reflecting seismic waves with period of one second. This requires the mass fraction of solid below the ICB to grow to an appreciable fraction in roughly one kilometer. This rapid growth of solid with depth is a result of downward fluid flow from the outer core which is a part of the convective motions which sustain the geodynamo. The solid which crystallizes from this descending fluid after it crosses the ICB continually coats the dendrites which occur there. The gradual cooling of the outer core causes the ICB to advance by growth of dendrites at their tips. The balance of these two effects gives an equilibrium profile for the mass fraction of solid with depth below the ICB which is capable of yielding sharp reflection of seismic waves.     

Long wavelength thermal convection between non-conducting boundaries

Non-linear Rayleigh-Bénard convection in a fluid layer is considered as a model of convection in the Earth's upper mantle. Previous studies have shown that when the temperature is held fixed at one of the boundaries of the layer, convection takes place in cells of width of the order of the layer depth or less. We investigate the effects of a different thermal boundary condition, in which the flux of heat is held fixed on both layer boundaries; then if this flux is just greater than that required for the onset of convection, motion takes place on horizontal scales much greater than the layer depth. An analytical treatment of the equations, based on an expansion in the depth-to-width ratio of the cells, shows that cells of a definite horizontal scale are the fastest growing according to linearised theory, but that these cells are unstable to ones of larger wavelength than themselves. Thus the dominant wavelength lengthens with time. The results hold whether the heat flux is generated internally of comes from beneath the layer. These results produce flow patterns similar to those found when the heat flux is much greater than the critical value. The results have important consequences for the understanding of mantle convection.     

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.     

A theoretical study of hydrothermal convection and the origin of the ophiolitic sulphide ore deposits of Cyprus

Hydrothermal circulation of seawater has been suggested as a mass transport mechanism for the formation of sulphide ore deposits in the ophiolitic rocks of Cyprus. Since ophiolitic sequences are generally regarded as fragments of oceanic crust and upper mantle, hydrothermal circulation of a form inferred from geological observations on Cyprus may be analogous to that thought to occur in oceanic crust at spreading ridges. The hypothesis that ore deposits were formed in ascending plumes of hot, buoyant fluid is examined by considering thermal convection in a permeable medium. To match the inferred pattern of circulation, finite amplitude convection in a cylindrical geometry is studied using finite difference approximations. These results combined with available geological and geochemical data are applied to understand better the physical controls on mineralisation.A simple model for the formation of the hydrothermal ore deposits of Cyprus is discussed. The model is semi-quantitatively reasonable in terms of vertical fluid flow rate, thermal structure, permeability and basal heat flow, and predicts volumes of maximum mineralisation similar to those observed. Three factors are identified which were important in confining mineralisation to a small volume immediately beneath the sea water/rock boundary: (1) hot fluid was confined to a narrow core zone of a rising plume, (2) the upward fluid flux was greatest in this same core zone, and (3) significant temperature decrease occurred within a thin surface boundary layer.     

Geoneutrinos and the energy budget of the Earth

The total energy loss of the Earth is well constrained by heat flux measurements on land, the plate cooling model for the oceans, and the buoyancy flux of hotspots. It amounts to 46 ± 2 TW. The main sources that balance the total energy loss are the radioactivity of the Earth's crust and mantle, the secular cooling of the Earth's mantle, and the energy loss from the core. Only the crustal radioactivity is well constrained. The uncertainty on each of the other components is larger than the uncertainty of the total heat loss. The mantle energy budget cannot be balanced by adding the best estimates of mantle radioactivity, secular cooling of the mantle, and heat flux from the core. Neutrino observatories in deep underground mines can detect antineutrinos emitted by the radioactivity of U and Th. Provided that the crustal contribution to the geoneutrino flux can be very precisely calculated, it will be possible to put robust constraints on mantle radioactivity and its contribution to the Earth's energy budget. Equally strong constraints could be obtained from a deep ocean observatory without the need of crustal correction. In the future, it may become possible to obtain directional information on the geoneutrino flux and to resolve radial variations in concentration of heat producing elements in the mantle.     

五矩二流太阳风等离子体特性的数值研究

本文数值求解了各向同性二流太阳风的五矩方程组,得出了1 R_s-2AU区域内太阳风密度、速度、电子和质子温度、它们的热流通量密度q以及非麦克斯韦分布尾部粒子过剩量ξ随日心距离的变化关系.文中比较了二流太阳风五矩模型、四矩模型（ξ=0）和低阶矩模型（不包括q和ξ二个矩方程）的等离子体特性,着重讨论了量ξ对质子温度及其热流通量的影响.结果表明,包括言的五矩方程可改善T_e/T_p和q_e/q_p的计算值与观测值的符合程度.     

Changes in the saltation flux following a step‐change in macro‐roughness

The effect of a step change in macro‐roughness on the saltation process under sediment supply limited conditions was examined in the atmospheric boundary layer. For an array of roughness elements of roughness density λ = 0.045 (λ = total element frontal area/total surface area of the array) the horizontal saltation flux was reduced by 90% (±7%) at a distance of ≈150 roughness element heights into the array. This matches the value predicted using an empirical design model and provides confidence that it can be effectively used to engineer roughness arrays to meet sand flux reduction targets. Measurements of the saltation flux characteristics in the vertical dimension, including: saltation layer decay (e‐folding) height and particle size, revealed that with increasing distance into the array, the rate of mass flux change with increasing height decreased notably, and (geometric) mean particle diameter decreased. The distribution of the saltation mass flux in the vertical remains exponential in form with increasing distance into the roughness array, and the e‐folding height increases as well as increasing at a greater rate as particle diameter diminishes. The increase in e‐folding height suggests the height of saltating particles is increasing along with their mean speed. This apparent increase in mean speed is likely due to the preferential removal, or sequestration, of the slower moving particles across the size spectrum, as they travel through the roughness array. Copyright © 2018 John Wiley & Sons, Ltd.     

A simple model of convection in the terrestrial planets

Abstract

In this paper we study analytically the simplest fluid mechanical model which can mimic the convective behavior which is thought to occur in the solid mantles of the terrestrial planets. The convecting materials are polycrystalline rocks, whose creep behavior depends very strongly on temperature and probably also on pressure. As a simple model of this situation, we consider the flow of a Newtonian viscous fluid, whose viscosity depends strongly on temperature (only), and in fact has an infinite viscosity below a certain temperature, and a constant viscosity above this temperature. This model would also be directly relevant to the convection of a melt beneath its own solid phase (e.g. water below ice, though in that case there are other physical complications).

As a consequence of this assumption, there is a vigorous convection zone overlain by a stagnant lid, as also observed in analogous laboratory experiments (Nataf and Richter, 1982). The analysis is then very similar to that of Roberts (1979), but the extension to variable viscosity introduces important differences, most notably that the boundary between the lid and the convecting zone is unknown, and not horizontal. The resulting buoyancy induced stresses near this boundary are much larger than the stresses produced by buoyancy in the side-wall plumes, and mean that the dynamics of this region, and hence also the heat flux, are independent of the rest of the cell. We give a first order approximation for the Nusselt number-Rayleigh number relationship.     

A note on the heat transfer by the axisymmetric thermal convection in a rotating fluid annulus

Abstract

Some new measurements are presented of the axisymmetric heat transport in a differentially heated rotating fluid annulus. Both rigid and free upper surface cases are studied, for Prandtl numbers of 7 and 45, from low to high rotation rates. The rigid lid case is extended to high rotation rates by suppressing the baroclinic waves, that would normally develop at some intermediate rotation rate, with the use of sloping endwalls.

A parameter P is defined as the square of the ratio of the (non-rotating) thermal sidewall layer thickness to the Ekman layer thickness. For small P the heat transport remains unaffected by the rotation, but as P increases to order unity the Ekman layer becomes thin enough to inhibit the radial mass transport, and hence the heat flux. No explicit Prandtl number dependence is observed. Also this scaling allows the identification of the region in which the azimuthal velocity reaches its maximum. Direct comparisons are drawn with previous experimental and numerical results, which show what can be interpreted as an inhibiting effect of increasing curvature on the heat transport.     

Particle accumulation in a cylindrical sediment trap under laminar and turbulent steady flow: An experimental approach

The hydrodynamical, fluid and particle parameters which control flushing rates, flow cells, and accumulation rates of particulate matter in cylindrical (MultiPIT) sediment traps were quantified in a flume simulation using a seeding technique for 25–45 µm particles. Particle collection was found to be a trap- and particle-specific filtering process encompassing advective and gravitational entry of particles over a reduced trap aperture area, and gravitational-turbulent removal of particles at the bottom of the internal flow cell. Trapping efficiency increased up to 10-fold with increasing horizontal flow velocity (1–30 cm · s^–1). For given flow velocity, the trap over-and undercollected particles relative to their weight, i.e. (theoretical) Stokes settling velocity. The trapping efficiency increased with increasing trap Reynolds number Re_T, changed by the approaching velocity in our experiments. Opposite findings from earlier experiments using the flume seeding technique and changing Re_T by altering the trap diameter (Butman, 1986) are discussed. Semi-empirical equations are derived for the accumulation process of light, heavy and intermediate particles. From these, measured trap fluxes can be converted into in-situ verticle particle flux except for light particles.