Top Thermal Boundary Layer Research Articles

Flows in horizontal thermohaline convection with differential diffusion

Oceanographers use the term “differential diffusion” to express a greater value of bulk turbulent diffusivity of temperature within the ocean than the value of bulk diffusivity of salinity, the ratio quantified by Lewis number. Investigation of horizontal thermohaline convection at Prandtl number 1 and infinity over the range reveals a variety of new flow patterns. The chamber has a linearly changing temperature T and salinity S along the top extending from the cold, fresh “polar” end to the hot, salty “tropics” end. It has an aspect ratio of 8 and sides and bottom are insulated and impermeable. Five transition flow patterns occur with little hysteresis for a fixed salinity Rayleigh number Ras of order 105 as Rayleigh number Ra changes from 3.2 × 106 down to 1. They are: 1. A steady T-cell with sinking at the cold end flowing into a bottom flow that feeds up into a top thermal boundary layer. 2. Salty blobs in the boundary layer that amplify and move from the hot to cold end. Each cold end arrival triggers a sudden increase in overturning velocity. 3. A “stripes” pattern where top to bottom cells (alternating T and S cells) move toward the cold end. 4. An S-cell that is a mirror image of the T-cell near the top along with small T-cells lying at the bottom that move toward the cold end 5. A steady S-cell. Each pattern has a distinct volumetric signature in a T-S diagram. Ranges of Ra with various patterns are sizeable at Ras=7.5 × 105 if Le >4/3 but insensitive to Pr. Balanced convection at Ra = Ras >106 adopts a large unsteady supercell containing smaller T and S cells. Exact ranges of the supercell are unknown. Since differential diffusion produces a large collection of flows compared to thermal convection alone, it might produce unexpected new results if added into numerical models of the ocean.

Planform of small-scale convection under the island arc

We have proposed an existence of small-scale convection in the mantle wedge in order to explain the alignment of group of volcanoes in the NE Honshu subduction zone, Japan. The roll type thermal convection whose axis is normal to the plate boundaries and “flip-flopping” of rolls (warmer and colder parts exchange their positions, as time passes) are reported. However, the subsequent study shows an existence of non-flip-flopping rolls, also. Thus, in this study, I investigate the possible causes of these differences systematically. I found that along-arc wavelength of small-scale convection is controlled by two length scales which may be associated with the inclined nature of the bottom of the low viscosity mantle wedge where the small-scale convection emerges. When the convection is in early stage of evolvement or the speed of subducting slab is small, the long-wavelength rolls become prominent. As the convection evolves or the speed of subduction increases, short-wavelength rolls tend to take over the long-wavelength rolls. The transition from the long-wavelength to the short-wavelength rolls occurs in a several way. It may occur through the splitting of rolls, adjustment of roll patterns and flip-flopping. If the pattern becomes short-wavelength roll, it becomes stable. The range of existence of flip-flopping is limited and the existence of viscosity jump in the top thermal boundary layer may be required.

Scaling of stagnant-lid convection with Arrhenius rheology and the effects of mantle melting

SUMMARY On the basis of numerical modelling and scaling analysis, a few modifications are proposed for the scaling of stagnant-lid convection, in order to make it more applicable to the thermal evolution of terrestrial planets. The effect of using Arrhenius rheology, as opposed to more popular linear-exponential rheology, is first investigated, and the stability analysis of top thermal boundary layer is shown to be able to capture systematic differences caused by the different kinds of temperature-dependent viscosity. The local stability analysis is then extended to handle the effects of mantle melting such as dehydration stiffening and compositional buoyancy. A new heat-flow scaling law incorporating these effects suggests that mantle melting may reduce the conventional prediction of surface heat flux by up to a factor of ∼5–10, and its potential impact on our understanding of planetary evolution is briefly discussed.

Quantitative investigation of physical properties of mantle plumes in three-dimensional numerical models

Numerical calculations have been carried out to investigate the physical properties of mantle plumes in highly viscous thermal convection depending on the Rayleigh number (Ra). The Boussinesq approximation was applied in a three-dimensional Cartesian domain filled with isoviscous, purely bottom-heated fluid with infinite Prandtl number. In order to monitor the dynamical behavior of plumes an automatic plume detecting routine was developed based on the temperature between the plume and its surroundings. It was established that as the convection becomes more vigorous with increasing Rayleigh number the average cross-sectional area of an individual plume decreases (∼Ra−2∕3), the vertical velocity in plumes increases (∼Ra2∕3), while the average temperature in plumes is independent of Ra. It means that the volume and the heat transport in an individual plume is independent of the Rayleigh number. The number of plumes forming in the box increases (∼Ra1∕3) which is in accordance with the scale analysis using the energy balance and the conservation of momentum. Furthermore, the Rayleigh number influences the temporal behavior of the average surface heat flow [Nusselt number, Nu0(t)] and the heat advected by plumes [Twp(t)]. The characteristic frequencies of Nu0(t) and Twp(t) increase by ∼Ra2∕3 in agreement with the rate of increase of the vertical velocity in plumes. The characteristic frequencies of Nu0(t) and Twp(t) are between the frequency corresponding to the time necessary for a plume to rise from the bottom to the top of the layer and the frequency of a whole convective cycle. The time series of Twp(t) contain larger amplitudes and higher frequencies than Nu0(t). It was assumed that the heat in the top thermal boundary layer (TBL) propagates by conduction and using Twp(t) as an input at the bottom of the top TBL the amplitude and the frequency of the heat flow series on the surface was calculated. It corresponds very well to the amplitude and the frequency of the observed Nu0(t). The correlation analysis between the time series of the surface Nusselt number and the heat advected by hot plumes showed that the time delay between the time series is equal to the time of the heat propagation by conduction through the TBL. The correlation between time series Twp(t) at different depths demonstrated well that the main heat transfer mechanism in plumes is advection.

Degree‐one mantle convection: Dependence on internal heating and temperature‐dependent rheology

Geophysical and geological observations suggest that a degree‐one mantle flow pattern, consisting of one upwelling and one downwelling, may have existed at some time in the mantles of Mars, the Moon, and perhaps even for the Earth during times of supercontinent formation. Simple fluid experiments utilizing isoviscous rheologies predict shorter wavelength flow patterns, and it is therefore important to determine fluid dynamical parameter sets which lead to larger wavelength flow patterns consistent with observations. We perform a series of numerical fluid dynamics calculations in a spherical 3‐D geometry in which we vary Rayleigh number, rheological activation parameter, and the degree of internal heating in order to define which parameter choices can lead to degree‐one mantle convection. We find that increasing the degree of internal heating increases the interior temperature of the mantle which leads to a larger viscosity contrast across the top thermal boundary layer, and that degree‐one mantle flow occurs only in cases which exhibit internal heating, utilize activation coefficients which lead to 103 or higher viscosity contrasts across the mantle, and have viscosity contrasts across the top thermal boundary layer within the range of 200–3000.

Delayed cooling of the Earth’s mantle due to variable thermal conductivity and the formation of a low conductivity zone

The thermal conductivity of mantle materials decreases with higher temperatures and increases with greater pressure due to phonon mechanism, but increases with temperature for radiative transfer. This trade-off attribute allows for the formation of a low conductivity zone (LCZ) within the top thermal boundary layer in mantle convection. We have studied with a two-dimensional (2-D) cartesian model the potential of this low conductivity zone in retarding secular cooling of the mantle for a longer period of time than models with constant thermal conductivity. Using a recently proposed model for thermal conductivity and an adiabatic boundary condition for the bottom of the mantle, we have carried out a set of numerical experiments within the framework of the extended Boussinesq approximation for constant viscosity, depth-dependent thermal expansivity and variable thermal conductivity with surface Rayleigh numbers between around 106 and around 107. We have employed internal heating between two and four times the chondritic level and half-life time values of 2.5 Ga and 5 Ga. The cooling rate of the mantle can be decreased by the feedback interaction between mantle conductivity and internal heating, which gives rise to slower sinking cold currents. The retardation time increases with the strength of radiogenic heating and can be as long as a couple of billion years for high initial heating rates, four times the chondritic value. However, increasing the radiative contribution of the conductivity speeds up the cooling process. The less the radiative contribution is, the more mantle cooling would be retarded. In the course of adiabatic cooling we find the paradoxical situation in which the effective Rayleigh number of the mantle can actually increase with time. We suggest then that, because of the possibilities for the formation of low conductivity zones adjacent to the thermal boundary layers, which bring forth unexpected consequences, mantle thermal conductivity should be considered as a fundamentally important factor in constraining the thermal evolution of both the core and mantle.

Convective instabilities induced in a critical fluid

We study temperature propagations and convective instabilities induced in critical fluids experimentally, theoretically and numerically. The analytical system is as follows: A critical fluid, the temperature of which is slightly higher than the critical temperature, is confined between two horizontal plates. The temperature of the bottom plate is raised at time zero and the temperature of the top plate is kept at the initial temperature of the fluid. An experimental system for the visualisation of convection induced in critical CO 2 is developed. In the theoretical analysis, we derive the thermofluiddynamics equations of critical fluids, in which the compressibility of the fluids and the pressure dependence of the entropy are taken into account. We solve the governing equations by the control volume finite difference method, in which the MacCormack scheme is employed for the time derivative, and investigate the temperature propagation modes on short, intermediate and long time scales. Through the analysis, the following results have been obtained; (1) On short time scales: Temperature propagates very quickly as acoustic waves because of the high compressibility of the critical fluids. (2) On intermediate time scales: The system temperature rises quickly and becomes constant in the central part of the fluid because of the quick temperature wave propagations. Thin thermal boundary layers are established near the bottom and top walls through the top wall is not cooled. (3) Long time scales: Thermal plumes are induced upwards from the bottom thermal boundary layer. Plumes are also initiated downwards from the top thermal boundary layer although the top wall temperature is not cooled. The convective mode of the thermal plumes driven in the critical fluids are different from that of Rayleigh-Bénard convection. Finally, we explain and discuss future problems related to transport phenomena in critical fluids.

A possible role of weak zone at plate margin on secular mantle cooling

We study a simple 2‐D model of secular mantle cooling by convection in which the viscosity depends exponentially on the temperature. To understand the effects of weak zones near the sinking and rising areas, their position and viscosity are imposed. The heat transport efficiency is measured by introducing local Rayleigh (Ral) and Nusselt (Nul) numbers defined at each thermal boundary layer. We find that the Nul ‐ Ral relationship for the bottom boundary layer is similar to that of the steady‐state convection with constant viscosity in a rectangular box. The existence of weak zones predicts a higher Nul for the top thermal boundary layer, suggesting that this will enhance the heat transport of the top layer. If we change Ral of the top layer into the higher of the two Rals, which is usually that of bottom layer, the resultant Nul ‐ Ral approaches that of steady‐state constant viscosity convection. This may imply that the heat transport of convection with secular cooling may be controlled by the boundary layer which has higher Ral. We also suggest that the commonly used parameterization, in which the convecting medium is treated as having a constant viscosity equal to that of its isothermal core, may overestimate the heat transport efficiency.

Effects of chemical fractionation of heat-producing elements on mantle evolution inferred from a numerical model of coupled magmatism-mantle convection system

A numerical model of a coupled magmatism-mantle convection system has been developed to study how a strong chemical fractionation of heat-producing elements as a result of mantle magmatism influences the evolution of the uppermost mantle. Mantle convection is modeled by a convection of a binary eutectic material A ξ B 1− ξ with a Newtonian rheology in a two-dimensional rectangular box; A stands for olivine and B stands for a mixture of pyroxene and garnet. The viscosity depends strongly on temperature and the density increases with decreasing ξ and temperature. The material contains decaying heat-producing elements. Mantle magmatism is modeled by an extraction of melt generated at depth by a pressure-release partial melting and a deposit of the extracted melt in a thin crustal layer at the top of the box. The melt extraction and deposit give rise to heterogeneity in the distribution of heat-producing elements as well as in ξ distribution. A dense, low-ξ magmatic product is more enriched in heat-producing elements than is a less dense, high-ξ residual material. The positive thermal buoyancy that a low-ξ material gains owing to its enrichment with heat-producing elements causes a catastrophic change in the chemical structure of the uppermost mantle with time when the viscosity in the top thermal boundary layer (lithosphere) is sufficiently high. A chemically stratified structure with a high-ξ residual material in the upper part and a low-ξ material in the lower part of the box develops at the beginning, when active magmatism occurs owing to strong internal heating. As the heat-producing elements decay and the magmatism becomes weaker, a steep gradient or a discontinuity in ξ distribution develops between the high-ξ part and the low-ξ part of the box, and the convection occurs temporarily as a layered convection. The chemically layered structure is, however, catastrophically destroyed and the box suddenly becomes more chemically homogeneous at a subsequent time when the positive thermal buoyancy of the low-ξ material at depth becomes stronger than the chemically induced negative buoyancy of the low-ξ material. The catastrophic change in chemical structure induces a drop in temperature at depth and a sudden enhancement of mantle magmatism. When the viscosity in the lithosphere is sufficiently low, in contrast, the chemically layered structure is stable and the catastrophic change from a chemically layered structure to a more chemically homogeneous structure does not occur. A catastrophic destruction of a chemically layered structure in the upper mantle as a result of strong fractionation of heat-producing elements may have occurred in the early Earth.

Perturbation analysis of convective instability of oceanic lithosphere and initiation of subduction zones

A linear perturbation analysis is made on convective instability of the oceanic lithosphere to understand the occurrence of subduction on the Earth. The oceanic lithosphere is modeled by the upper part of top thermal boundary layer in a convecting Newtonian temperature‐dependent viscosity fluid. Thickening of oceanic lithosphere with its age due to cooling at the surface is modeled by thickening of the thermal boundary layer with time; its thickening rate is a measure of the rate with which temperature in the thermal boundary layer decreases with time and hence is a measure of the rate with which viscosity in the thermal boundary layer increases with time. The linear perturbation analysis shows the possible excitation of plural eigenmodes to give rise to an instability in the thermal boundary layer depending on a threshold value Rc of the local Rayleigh number R of the thermal boundary layer. Further, Rc is shown to be a linearly increasing function of , which is treated as a free parameter. From estimates of R and appropriate for the major oceanic lithosphere (denoted as Robs and obs, respectively), I find that Rc( = 0) < Robs < Rc( = obs). The flow pattern of mantle convection depends strongly on which eigenmodes are excited by the convective instability in thermal boundary layer: (1) When R is lower thanRc, only the fundamental and the first higher modes at most are excited. The resulting mantle convection manifests itself as the secondary convection. (2) When R is higher than Rc, the modes higher than the first higher mode are also excited. The resulting mantle convection involves lithospheric instability, leading to large local deformation in the oceanic lithosphere and finally to subduction of the oceanic lithosphere. This result implies that (1) subduction does not begin newly in the major oceanic lithosphere at present, since lithospheric instability is suppressed by thickening of the major oceanic lithosphere, neither by its high viscosity nor by its buoyancy, (2) subduction will newly begin in the major oceanic lithosphere due to lithospheric instability in future when the oceanic lithosphere becomes so old that its thickening rate decreases to sufficiently small value. The latter result suggests that lithospheric instability is an important agent that maintains multiplate tectonics on the Earth.