Numerical models of mantle convection with secular cooling

A 2-D time-dependent finite-difference numerical model is used to investigate the thermal character and evolution of a convecting layer which is cooling as it convects. Two basic cooling modes are considered: in the first, both upper and lower boundaries are cooled at the same rate, while maintaining the same temperature difference across the layer; in the second, the lower boundary temperature decreases with time while the upper boundary temperature is fixed at 0°C. The first cooling mode simulates the effects of internal heating while the second simulates planetary cooling as mantle convection extracts heat from, and thereby cools, the Earth's core. The mathematical analogue between the effects of cooling and internal heating is verified for finite-amplitude convection. It is found that after an initial transient period the central core of a steady but vigorous convection cell cools at a constant rate which is governed by the rate of cooling of the boundaries and the viscosity structure of the layer. For upper-mantle models the transient stage lasts for about 30 per cent of the age of the Earth, while for the whole mantle it lasts for longer than the age of the Earth. Consequently, in our models the bulk cooling of the mantle lags behind the cooling of the core-mantle boundary. Models with temperature-dependent viscosity are found to cool in the same manner as models with depth-dependent viscosity; the rate of cooling is controlled primarily by the horizontally averaged variation of viscosity with depth. If the Earth's mantle cools in a similar fashion, secular cooling of the planet may be insensitive to lateral variations of viscosity.     

Time-dependent thermal convection, mantle differentiation and continental-crust growth

The thermal evolution of the Earth is controlled by radioactive elements whose heat production rate decays with time and whose spatial distribution depends on chemical segregation processes.
We present a 2-D and finite-difference Boussinesq convection model with temperature-dependent viscosity and time- and space-dependent radioactive heat sources. We used Newtonian rheology, boxes of aspect ratio 3, and heating from within. Starting from the geochemical results of Hofmann (1988), it is assumed that the radioactive heat sources of the mantle were initially distributed homogeneously. In a number of calculations, however, higher starting abundances of radioactive sources were assumed in the upper mantle. For the present geological situation, this also results in a depleted upper mantle. It was assumed that, if the viscosity falls below a certain critical value, chemical segregation will take place. In this way, model continental crust develops, leaving behind areas of a depleted mantle. We obtained the heat source, flow line, temperature, viscosity and heat-flow distribution as a function of time with realistic values, especially for the present time. The present viscosity of the upper mantle is approximately at the standard value obtained for postglacial uplift modelling; the deeper-mantle viscosity is considerably higher. The time dependence of the computed mean of the kinetic energy of mantle convection bears a resemblance to that of the magmatic and orogenetic activity of the Earth. We assumed that the 670 km discontinuity cannot be penetrated by the flow.     

A boundary layer model for mantle convection with surface plates

Summary. A simple, analytical model for mantle convection with mobile surface plates is presented. Our aim is to determine under what conditions free convection can account for the observed plate motions, and to evaluate the thermal structure of the mantle existing under these conditions. Boundary layer methods are used to represent two-dimensional cellular convection at large Rayleigh and infinite Prandtl numbers. The steady-state structure consists of cells with isentropic interiors enclosed by thermal boundary layers. Lithospheric plates are represented as upper surfaces on each cell free to move at a uniform speed. Buoyancy forces are concentrated in narrow rising and decending thermal plumes; torques imparted by these plumes drive both the deformable mantle and overlying plate. Solutions are found for a comprehensive range of cell sizes. We derive an expression for the plate speed as a function of its length, the mantle viscosity and surface heat flux. Using mean values for these parameters, we find that thermal convection extending to 700 km depth can move plates at 1 cm yr^-1, while convection through the whole mantle can move plates at 4–5 cm yr^-1. Analysis of the steady-state temperature field, for the case of heating from below, shows that the upper thermal boundary layer develops a complex structure, including an 'asthenosphere' defined by a local maximum in the geotherm occurring at depths of 50–150 km.     

Plate generation and two-phase damage theory in a model of mantle convection

The formation of narrow, rapidly deforming plate boundaries separating strong plate interiors are integral components of the generation of plate tectonics from mantle convection. The development of narrow plate boundaries requires the interaction of a non-linear rheology and convection. One such non-linear rheology is two-phase damage theory which employs a non-equilibrium relation between interfacial surface energy, pressure and viscous deformation, thereby forming a theoretical model for void generation. Two-phase damage theory was recently extended to allow for deformational work to increase the fineness (reduce the grain size) of the matrix phase. We present results testing two-phase damage theory in a 2-D convectively driven system where we allow for (1) pure void-generating damage, (2) pure fineness-generating damage and (3) combined void- and fineness-generating damage. Pure void-generating damage is found to be unsuccessful at producing plate-like features. Fineness-generating damage is successful at inducing plate-like behaviour in certain circumstances, including increasing viscosity sensitivity to fineness and certain regimes of damage input and healing rate. Cases with combined void- and fineness-generating damage produce significantly more localization than the end-members due to the apparent increase of deformational work input into fineness generation. The interaction of microcracks and grain size reduction in two-phase damage theory suggests a rheological model for shear localization necessary for the formation of plate tectonic boundaries.     

On the existence of a second scale of convection in the upper mantle

Summary. This paper is concerned with an examination of the possibility that there might exist a small scale of convective circulation beneath the oceanic lithosphere. Recent suggestions that this might be the case have been made in an effort to understand why the bathymetry of the sea-floor deviates from the prediction of boundary layer theory for ages in excess of about 100 Ma. The energy source which sustains the secondary motion is supposed to be found in the steep temperature gradient near the planetary surface which is itself presumably maintained by the large-scale convective circulation associated with plate creation and destruction. Here we investigate the extent to which the temperature dependence of viscosity may act so as to stabilize the upper boundary layer against disruption by such secondary instability. If the viscosity profile is monotonie and the asymptotic upper mantle viscosity is about 10²²poise, as suggested by post-glacial rebound data, then the existence of the second scale is extremely unlikely. On the other hand, if a sufficiently pronounced low viscosity zone does exist under old sea-floor then the development of such a second scale cannot be ruled out completely. Some recently obtained geophysical evidence is reviewed which suggests that this is unlikely to be the case.     

Eigendecomposition of the two-point correlation tensor for optimal characterization of mantle convection

The two-point correlation tensor provides complete information on mantle convection accurate up to second-order statistics. Unfortunately, the two-point spatial correlation tensor is in general a data-intensive quantity. In the case of mantle convection, a simplified representation of the two-point spatial correlation tensor can be obtained by using spherical symmetry. The two-point correlation can be expressed in terms of a planar correlation tensor, which reduces the correlation's dependence to only three independent variables: the radial locations of the two points and their angular separation. The eigendecomposition of the planar correlation tensor provides a rational methodology for further representing the second-order statistics contained within the two-point correlation in a compact manner. As an illustration, results on the planar correlation are presented for the thermal anomaly obtained from the tomographic model of Su, Woodward & Dziewonski (1994 ) and the corresponding velocity field obtained from a simple constant-viscosity convection model Zhang & Christensen 1993 ). The first 10 most energetic eigensolutions of the planar correlation, which constitute an almost three orders of magnitude reduction in the data, capture the two-point correlation to 97 per cent accuracy. Furthermore, the energetic eigenfunctions efficiently characterize the thermal and flow structures of the mantle. The signature of the transition zone is clearly evident in the most energetic temperature eigenfunction, which clearly shows a reversal of thermal fluctuations at a depth of around 830  km. In addition, a local peak in the thermal fluctuations can be observed around a depth of 600  km. In contrast, due to the simplicity of the convection model employed, the velocity eigenfunctions exhibit a simple cellular structure that extends over the entire depth of the mantle and do not exhibit transition-zone signatures.