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.     

Convection with heat flux prescribed on the boundaries of the system. I. The effect of temperature dependence of material properties

Abstract

We study the bifurcation to steady two-dimensional convection with the heat flux prescribed on the fluid boundaries. The fluid is weakly non-Boussinesq on account of a slight temperature dependence of its material properties. Using expansions in the spirit of shallow water theory based on the preference for large horizontal scales in fixed flux convection, we derive an evolution equation for the horizontal structure of convective cells. In the steady state, this reduces to a simple nonlinear ordinary differential equation. When the horizontal scales of the cells exceed a certain critical size, the bifurcation to steady convection is subcritical and the degree of subcriticality increases with increasing cell size.     

Linear Stability of Thermal Convection in Rotating Systems with Fixed Heat Flux Boundaries

Linear stability of rotating thermal convection in a horizontal layer of Boussinesq fluid under the fixed heat flux boundary condition is examined by the use of a vertically truncated system up to wavenumber one. When the rotation axis is in the vertical direction, the asymptotic behavior of the critical convection for large rotation rates is almost the same as that under the fixed temperature boundary condition. However, when the rotation axis is horizontal and the lateral boundaries are inclined, the mode with zero horizontal wavenumber remains as the critical mode regardless of the rotation rate. The neutral curve has another local minimum at a nonzero horizontal wavenumber, whose asymptotic behavior coincides with the critical mode under the fixed temperature condition. The difference of the critical horizontal wavenumber between those two geometries is qualitatively understood by the difference of wave characteristics; inertial waves and Rossby waves, respectively.     

Destabilization of a 650 km chemical boundary layer and its bearing on the evolution of the continental crust

A mechanism for the production of a chemical change in the mantle, from primordial silicate compositions above the 650 km discontinuity to differentiated compositions below, is reviewed. Some consequences of this are the stabilization of two layer convection with a temperature contrast between the anhydrous mantle solidus and the geotherm which, at 650 km depth, is lower than any other location in the mantle. With thermal contributions from the concentration of the heat producing elements U, Th and K below the 650 km mantle boundary layer and the higher geotherms in the past, widespread or catastrophic melting may have taken place at this location. An episodic breach of this boundary layer by extensive heat and mass transport may have periodically destroyed any simple two-layer convection geometry in the mantle. Such episodic injections of mass and energy into the upper mantle from below may have been the mechanism responsible for episodes of enhanced surface tectonism and thermal activity which appear to be recorded in apparent polar wandering paths and radiometric ages of continental rocks.     

On the aspect ratios of two-layer mantle convection

Some dynamic implications of separate convection systems in the upper and the lower mantle of the Earth are investigated. It is shown that the horizontal scale of convection cells in the lower mantle is likely to determine the scale of flow in the upper mantle. This does not preclude the nearly independent realization of convection cells with horizontal dimensions comparable to the depth of the upper mantle.     

Convective instabilities in a variable viscosity fluid cooled from above

Whether in the mantle or in magma chambers, convective flows are characterized by large variations of viscosity. We study the influence of the viscosity structure on the development of convective instabilities in a viscous fluid which is cooled from above. The upper and lower boundaries of the fluid are stress-free. A viscosity dependence with depth of the form ν₀ + ν₁ exp(?γ.z) is assumed. After the temperature of the top boundary is lowered, velocity and temperature perturbations are followed numerically until convective breakdown occurs. Viscosity contrasts of up to 10⁷ and Rayleigh numbers of up to 10⁸ are studied.For intermediate viscosity contrasts (around 10³), convective breakdown is characterized by the almost simultaneous appearance of two modes of instability. One involves the whole fluid layer, has a large horizontal wavelength (several times the layer depth) and exhibits plate-like behaviour. The other mode has a much smaller wavelength and develops below a rigid lid. The “whole layer” mode dominates for small viscosity contrasts but is suppressed by viscous dissipation at large viscosity contrasts.For the “rigid lid” mode, we emphasize that it is the form of the viscosity variation which determines the instability. For steep viscosity profiles, convective flow does not penetrate deeply in the viscous region and only weak convection develops. We propose a simple method to define the rigid lid thickness. We are thus able to compute the true depth extent and the effective driving temperature difference of convective flow. Because viscosity contrasts in the convecting region do not exceed 100, simple scaling arguments are sufficient to describe the instability. The critical wavelength is proportional to the thickness of the thermal boundary layer below the rigid lid. Convection occurs when a Rayleigh number defined locally exceeds a critical value of 160–200. Finally, we show that a local Rayleigh number can be computed at any depth in the fluid and that convection develops below depth z_r (the rigid lid thickness) such that this number is maximum.The simple similarity laws are applied to the upper mantle beneath oceans and yield estimates of 5 × 10¹⁵?5 × 10¹⁶ m² s^?1 for viscosity in the thermal boundary layer below the plate.     

Topographic forcing of supercritical convection in a porous medium such as the oceanic crust

We have performed laboratory experiments using a Hele-Shaw cell to model a saturated, porous layer with various sinusoidal upper boundaries. Our intent was to determine the range of conditions over which boundary topography can control the pattern of thermal convection within a porous layer, and thereby take the first step toward understanding why heat flow seems correlated with hypsography in many areas of the ocean floor.These experiments indicate that above the critical Rayleigh number, topography does not control the convection pattern, except when the topographic wavelength is comparable to the depth of water penetration. Scaled to the depth of the layer, the convective wavenumbers are restricted to values between 2.5 and 4.8—a range which brackets π, the natural wavenumber for convection in a porous slab with planar, isothermal, impermeable boundaries. Topographies within this range control the circulation pattern perfectly, with downwelling under valleys and upwelling aligned with topographic highs. Other topographies do not force the pattern, although in some cases, the convection wavenumber may be a harmonic of the topographic wavenumber. Unforced circulation cells wander and vary in size, because they are not locked to the topography.For these experiments we employed eight different topographies with non-dimensional wavenumbers between 1.43 and 8.17, and we studied the flow at Rayleigh numbers between zero and five times the critical Rayleigh number. The amplitude of each topography tapered linearly (over a factor of three to six) from one end of the apparatus to the other, and the mean topographic amplitude was 0.05 times the depth of the layer. Under these conditions, amplitude has only a minor effect on the structural form and vigor of supercritical convection.Our results may apply to submarine geothermal systems, sealed by a thin layer of impermeable sediment draped over the basement topography. In this case, the convection wavelength—as measured perhaps by the spatial periodicity of conductive heat flow—may be a good measure of the depth to which water penetrates the crust. Where the circulation correlates with the bottom topography, it may be because the topographic wavelength is comparable to the depth to which water penetrates the porous crust.     

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.     

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.     

The effect of radiative heat loss on steady cellular convection

Summary In the atmosphere there may be layers undergoing cellular convection with a much larger heat flux through the base of the layer than through the top. This may be either because there is a steady loss of heat by radiation from the body of the fluid or because the temperature is everywhere rising. In this latter case the temperature gradients could remain constant so that the mechanics would be the same as if the heat were being lost and the temperature kept steady. The fluid is considered incompressible as in the classical theory of cellular convection, and we determine the critical Rayleigh number for the onset of convection and the width to height ratio of the cells as functions of the heat loss. The problem, is in some respects analogous to that of the motion of a viscous fluid between rotating cylinders but in this case there are two non-dimensional-numbers-the Rayleigh number (g h ⁴/K v) and a number representing the ratio of the heat loss by radiation to the heat flux. It is found that the critical Rayleigh number is decreased and the cells widened as had already been found for the case of a fluid with transfer coefficients having a spatial variation, with free boundaries, but the cells are made more narrow if the boundaries are rigid.     

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.     

The structure of 2D mantle convection and stress fields: Effects of viscosity distribution

Spatial fields of temperature, velocity, overlithostatic pressure, and horizontal stresses in the Earth’s mantle are studied in two-dimensional (2D) numerical Cartesian models of mantle convection with variable viscosity. The calculations are carried out for three different patterns of the viscosity distribution in the mantle: (a) an isoviscous model, (b) a four-layer viscosity model, and (c) a temperature- and pressure-dependent viscosity model. The pattern of flows, the stresses, and the surface heat flow are strongly controlled by the viscosity distribution. This is connected with the formation of a cold highly viscous layer on the surface, which is analogous to the oceanic lithosphere and impedes the heat transfer. For the Rayleigh number Ra = 10⁷, the Nusselt number, which characterizes the heat transfer, is Nu = 34, 28, and 15 in models with constant, four-layered, and p, T-dependent viscosity, respectively. In all three models, the values of overlithostatic pressure and horizontal stresses σ _xx in a vast central region of the mantle, which occupies the bulk of the entire volume of the computation domain, are approximately similar, varying within ±5 MPa (±50 bar). This follows from the fact that the dimensionless mantle viscosity averaged over volume is almost similar in all these models. In the case of temperature- and pressure-dependent viscosity, the overlithostatic pressure and stress σ _xx fields exhibit much stronger concentration towards the horizontal boundaries of the computation domain compared to the isoviscous model. This effect occurs because the upwellings and downwellings in a highly viscous region experience strong variations in both amplitude and direction of flow velocity near the horizontal boundaries. In the models considered with the parameters used, the stresses in the upper and lower mantle are approximately identical, that is, there is no denser concentration of stresses in the upper or lower mantle. In contrast to the overlithostatic pressure field, the fields of horizontal stresses σ _xx in all models do not exhibit deep roots of highly viscous downwelling flows.     

On the geodynamic setting of kimberlite genesis

The emplacement of kimberlites in the North American and African continents since the early Palaeozoic appears to have occurred during periods of relatively slow motion of these continents. The distribution of kimberlites in time may reflect the global pattern of convection, which forces individual plates to move faster or slower at different times. Two-dimensional numerical experiments on a convecting layer with a moving upper boundary show two different regimes: in the first, when the upper boundary velocity is high, heat is transferred by the large-scale circulation and in the second, when the upper boundary velocity is lower, heat is predominantly transferred by thermal plumes rising from the lower boundary layer. For a reasonable mantle solidus, this second regime can give rise to partial melting beneath the moving plate, far from the plate boundaries. The transition between these modes takes place over a small range of plate velocities; for a Rayleigh number of 10⁶ it occurs around 20 mm yr^?1. We suggest that the generation of kimberlite magmas may result from thermal plumes incident on the base of a slowly moving plate.     

Thermal evolution of Venus

The theory of three-dimensional and finite-amplitude convection in a viscous spherical shell with temperature and pressure dependent physical parameters is developed on the basis of a modified Boussinesq fluid assumption. The lateral dependences of the variables are resolved through their spherical harmonic representations, whereas their radial and time dependences are determined by numerical procedures. The theory is then applied to produce thermal evolution models for Venus. The emphasis is on illustrating the effects of certain physical parameters on the thermal evolution rather than proposing a specific thermal history for the planet. The main conclusions achieved in this paper are (1) a significant portion of the present temperature in the mantle and heat flux at the surface of Venus is probably owing to the decay of a high temperature established in the planet at the completion of its core formation, (2) the effective Rayleigh number of the mantle is so high that even the lower order modes of convection cool the planet sufficiently and maintain an almost adiabatic temperature gradient in the convecting region and high temperature gradients in the thermal boundary layers, (3) the convection is oscillatory with avalanche type properties which induces oscillatory features to the surface heat flux and the thickness of the crustal layer, and (4) a planetary model with a recycling crust cools much faster than those with a permanently buoyant crust.The models presented in this paper suggest that Venus has been highly convective during its history until ～ 0.5 Ga ago. The vigorous convection was bringing hot and fresh material from the deep interior to the surface and dragging down the crustal slags, floating on the surface, in to the mantle. The rate of cooling of the planet was so high that its core has solidified. In the last 0.5 Ga the vigour of convection diminished considerably and the crustal slags developed into a global and permanently buoyant crustal layer. The tectonic style on Venus has, consequently, changed from the recycling of crustal plates to hot spot volcanics. At the present time the planet is completely solid, except in the upper part of its mantle where partial melting may occur.     

A gradational density model for gravity anomalies over oceanic ridges

The long-wavelength gravity anomalies observed over oceanic ridges have been interpreted in terms of horizontal slabs with lateral variation of density. The location of such a slab in the earth's interior is estimated to be between the depths of 350 and 430 km, which defines the boundaries of the upper phase-transition zone of the mantle. A total density contrast, between the end planes of the horizontal slab, of 0.3 g/cm³ appears to be satisfactory for the interpretation. This remarkable coincidence in depth and density contrast associated with the pyroxene-garnet transformation process is considered to suggest that this process may possibly be: (1) taking place laterally; and (2) generating the gradational density contact which is reflected in the gravity anomalies. In turn, the mechanism for this lateral phase transformation may ultimately be attributable to the convection currents in the asthenosphere.     

Stability of the boundary layer between the lithosphere and convecting mantle and the steady-state lithospheric geotherm

In the steady state, the convective boundary layer (CBL) (the transition from the lithosphere to the convecting mantle, the lithosphere-asthenosphere boundary) is on the verge of stability. This determines its depth, thickness, and the steady-state temperature distribution in the lithosphere. Had the mantle been homogeneous, the base of the lithosphere at the current potential temperature would lie globally at the same depth H _rh of 50 to 70 km. Actually, the regime of interaction of the mantle convection with the lithosphere is determined by the relationship between this depth and the thickness H _depl of the chemical boundary layer including the crust and the layer of the depleted rock. If the thickness of the chemical boundary layer is small H _depl < H _rh, as it is the case in the present-day oceanic mantle, the suboceanic regime is established with the mantle convection that does not reach the base of the chemical boundary layer. In this case, the top of CBL is located at depth H _rh, while the oceanic heat flow and the depth of the seafloor only depend on the potential temperature T _p and, within the areas where the crust is older than 60 to 70 Ma, are the same everywhere far from the disturbed territories (the hot points and the subduction zones). The absence of noticeable distinctions between the heat flow in the different oceanic basins suggests a global constancy of the potential temperature. If H _depl > H _rh, the subcontinental regime of the interaction of the mantle convection with the lithosphere is established. In this case, the CBL is immediately adjacent to the depleted lithosphere, its top is located at depth H _depl, and the surface heat flow q(T _p, H _depl) not only depends on the potential temperature T _p but also on the the thickness of the depleted lithosphere H _depl; it decreases with increasing H _depl and, therefore, with the age of the lithosphere. Given the potential temperature, the dependence q(T _p, H _depl) agrees well with the envelope of the results of kimberlite xenolith thermobarometry presented in the diagram of the deepest xenolith depth as a function of the heat flow. It is likely that in the lowest part of the continental lithosphere there is a zone of horizontal shear deformation, from where kimberlites entrain the strongly deformed and, at the same time, the deepest xenoliths. Besides, the azimuthal anisotropy of seismic velocities can be associated with this zone. The change in its direction with depth can be observed as the Lehmann discontinuity.     

Intermittent convection

Abstract

A theoretical analysis of pseudo two-dimensional, finite-amplitude, thermal convection is made for an infinite Prandtl number fluid which is subjected to a constant heat flux out of the top boundary and insulated at the bottom. For large Rayleigh numbers the convective flow becomes intermittent and the system is characterized by the following cyclic process: the formation of a thermal boundary layer by diffusion, the instability of this layer when it becomes sufficiently thick, the destruction of the layer by the convective flow, the dying down of the convection, and the reforming of the thermal boundary layer by diffusion. The periodicity and the horizontal wave number of the intermittent convective flow are found to be independent of the depth of the fluid layer but depend on the rate of cooling and the properties of the fluid.     

Nonlinear convection in an imposed horizontal magnetic field

Abstract

Nonlinear two-dimensional magnetoconvection, with a Boussinesq fluid driven across the field-lines, is taken as a model for giant-cell convection in the sun and late-type stars. A series of numerical experiments shows the sensitivity of the horizontal scale of convection to the applied field and to the Rayleigh number R. Overstable oscillations occur in cells as broad as they are deep, but increasing R leads to steady motions of much greater wavelength. Purely geometrical effects can cause oscillation: this work implies that strong horizontal field will in general lead to time-dependent convection.     

Laboratory experiments in a baroclinic annulus with heating and cooling on the horizontal boundaries

Abstract

Experiments have been performed in a cylindrical annulus with horizontal temperature gradients imposed upon the horizontal boundaries and in which the vertical depth was smaller than the width of the annulus. Qualitative observations were made by the use of small, suspended, reflective flakes in the liquid (water).

Four basic regimes of flow were observed: (1) axisymmetric flow, (2) deep cellular convection, (3) boundary layer convective rolls, and (4) baroclinic waves. In some cases there was a mix of baroclinic and convective instabilities present. As a “mean” interior Richardson number was decreased from a value greater than unity to one less than zero, axisymmetric baroclinic instability of the Solberg type was never observed. Rather, the transition was from non-axisymmetric baroclinic waves, to a mix of baroclinic and convective instability, to irregular cellular convection.     

On the advection of sharp material interfaces in geodynamic problems: entrainment of the D″ layer

The D″ layer is a dense and chemically distinct layer at the base of the convecting mantle. Numerical modeling of the entrainment of this layer by mantle convection requires the solution of the advective transport equation without introducing numerical diffusion across sharp material boundaries. We use our improved second moment numerical method to solve the equation. The method conserves the amount of material and the first and second moments of material distribution in each control volume. We first consider two examples of isothermal Rayleigh–Taylor instability to illustrate the performance of our method by comparing our results with those of a number of field, tracer and marker chain methods. We show that the performance of our method in minimizing the numerical diffusion is better than the field methods and comparable to the tracer and marker chain methods. We then study the instability of the dense D″ layer and its interaction with the overlying mantle. A range of density contrast between the D″ layer and the mantle, layer thickness, and the Rayleigh number, Ra, is examined. We show that for higher values of these parameters, the amount of entrainment decreases and the layer remains stable over longer periods of time. For very thick D″ layers and high Ra values, internal convection can take place within the layer.