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.     

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.     

Free-surface formulation of mantle convection—II. Implication for subduction-zone observables

Viscous and viscoelastic models for a subduction zone with a faulted lithosphere and internal buoyancy can self-consistently and simultaneously predict long-wavelength geoid highs over slabs, short-wavelength gravity lows over trenches, trench-forebulge morphology, and explain the high apparent strength of oceanic lithosphere in trench environments. The models use two different free-surface formulations of buoyancy-driven flows (see, for example, Part I): Lagrangian viscoelastic and pseudo-free-surface viscous formulations. The lower mantle must be stronger than the upper in order to obtain geoid highs at long wavelengths. Trenches are a simple consequence of the negative buoyancy of slabs and a large thrust fault, decoupling the overriding from underthrusting plates. The lower oceanic lithosphere must have a viscosity of less than to²⁴ Pa s in order to be consistent with the flexural wavelength of forebulges. Forebulges are dynamically maintained by viscous flow in the lower lithosphere and mantle, and give rise to apparently stiffer oceanic lithosphere at trenches. With purely viscous models using a pseudo-free-surface formulation, we find that viscous relaxation of oceanic lithosphere, in the presence of rapid trench rollback, leads to wider and shallower back-arc basins when compared to cases without viscous relaxation. Moreover, in agreement with earlier studies, the stresses necessary to generate forebulges are small (∼ 100 bars) compared to the unrealistically high stresses needed in classic thin elastic plate models.     

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.