Plate tectonics provides a remarkably accurate kinematic description of the motion of the earth's crust but a fully dynamical theory requires an understanding of convection in the mantle. Thus the properties of plates and of the mantle must be related to a systematic study of convection. This paper reviews both the geophysical information and the fluid dynamics of convection in a Boussinesq fluid of infinite Prandtl number. Numerical experiments have been carried out on several simple two-dimensional models, in which convection is driven by imposed horizontal temperature gradients or else by heating either internally or from below. The results are presented and analysed in terms of simple physical models. Although the computations are highly idealized and omit variation of viscosity and other major features of mantle convection, they can be related to geophysical measurements. In particular, the external gravity field depends on changes in surface elevation; this suggests an observational means of investigating convection in the upper mantle.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112074000784

Convection in the earth's mantle: towards a numerical simulation

In this paper we show that it is probable that at stresses greater than about 10−2 bar creep in the earth's mantle is caused by dislocation motion rather than by the mass transport of atoms through diffusion from one grain boundary to another. The latter process leads to Nabarro-Herring creep. Only at very low stresses (<10−2 bar) should Nabarro-Herring creep be dominant. Dislocation creep equations (which are nonlinear in stress) are used to estimate the ‘effective viscosity,’ at a constant creep rate of 10−16 sec−1, of the moon, Mars, and the mantles of Venus and the earth. The effective viscosity of the mantles of Venus and the earth diminishes with increasing depth, goes through a minimum, and then increases continuously up to the mantle-core boundary. The effective viscosity in the lower mantle is much smaller than estimates derived from the Nabarro-Herring creep equation. The effective viscosity in the deep interior of the moon is approximately constant and does not increase appreciable with depth. The effective viscosity of Mars is intermediate in behavior between that of the earth and of the moon.

The creep strength of the Earth's mantle

The factors affecting the thermal behavior of a lithospheric slab descending into the mantle are so numerous and complicated that only numerical methods can accurately account for them. A series of finite-difference calculations of the temperature field, including the effects of viscous dissipation, adiabatic compression, phase changes and radioactive heat generation are carried out, and the relative effects of different parameters are investigated. An analysis of the stability and convergence of the numerical method indicates that the errors are small and can be reduced to any desired level by varying the grid size and the time steps. At a crustal spreading rate of 8 cm/yr, with all heat sources, the slab reaches thermal equilibrium with the surrounding mantle at a depth of about 650 km. Among observable geophysical quantities, seismic travel times and amplitudes provide the most information about the slab. Surface heat flow is sensitive to subsurface conditions that are at relatively shallow depths, and gravity anomalies are broad and are masked by crustal effects. Three dimensional calculations predict strong bending of seismic rays near slabs, which causes strong focusing and produces shadow zones. Analysis of travel-time data for the Tonga-Fiji region indicates that waves propagating down the slab from shallow events are advanced by about 4 sec. Observations and theoretical travel-time anomalies based on calculated temperature fields are in general agreement.

Temperature Field and Geophysical Effects of a Downgoing Slab

Summary

The interaction of solid-solid phase transitions with convection in the Earth's mantle involves, for univariant systems: (1) effects of latent heat and advection of ambient temperature on the position of the phase boundary and on its associated body force, and (2) the coupling of latent heat with the ordinary thermal expansivity of the material. For divariant systems, an effective thermal expansion coefficient and a modified adiabatic temperature gradient may be defined for the phase transition zone. Linear stability theory for a fluid layer with a univariant phase change is reviewed and applied to the endothermic spinel-oxide transformation. The theory of the stability of a fluid layer with a divariant phase transformation is developed and critical Rayleigh numbers are given for a model of the olivine-spinel transition. Of special interest is the case where the Earth's temperature gradient exceeds the adiabatic temperature gradient outside the phase transition zone but is smaller than the increased adiabatic temperature gradient in the two-phase olivine-spinel region. The thermal structure of the descending lithosphere is calculated, including the effects of frictional heating on the slip zone and of the olivine-spinel and spinel-oxide transitions; temperature contrasts of 700 °K can exist between the slab and adjacent mantle at 800 km depth. The net body force on the descending slab due to thermal contraction and the major mineralogical phase changes is downward. The olivine-spinel transition may be responsible for the tensional focal mechanisms of intermediate depth earthquakes while the spinel-oxide transformation may be related to the compressional focal mechanisms of deep earthquakes.

/pdf/role-of-phase-transitions-in-a-dynamic-mantle-3jyo8wb7i9.pdf

Role of Phase Transitions in a Dynamic Mantle

In regions of slowly varying lateral density changes, the gravity and geoid anomalies may be expressed as power series expansions in topography. Geoid anomalies in isostatically compensated regions can be directly related to the local dipole moment of the density-depth distribution. This relationship is used to obtain theoretical geoid anomalies for different models of isostatic compensation. The classical Pratt and Airy models give geoid height-elevation relationships differing in functional form but predicting geoid anomalies of comparable magnitude. The thermal cooling model explaining ocean floor subsidence away from mid-ocean ridges predicts a linear age-geoid height relationship of 0.16 m/m.y. Geos 3 altimetry profiles were examined to test these theoretical relationships. A profile over the mid-Atlantic ridge is closely matched by the geoid curve derived from the thermal cooling model. The observed geoid anomaly over the Atlantic margin of North America can be explained by Airy compensation. The relation between geoid anomaly and bathymetry across the Bermuda Swell is consistent with Pratt compensation with a 100-km depth of compensation.

On isostatic geoid anomalies

The aspects of mantle convection that do not require a knowledge of the viscosity are considered first. Using spreading rates obtained from magnetic anomaly data, the temperature distribution in the thermal boundary layers adjacent to the ocean floors is determined. The values of the surface heat flux are compared with measurements. A temperature distribution in the upper mantle is obtained by matching the boundary-layer profiles to adiabatic profiles consistent with the boundary-layer theory for finite amplitude convection. Mechanisms for a fluid-like mantle are considered. Assuming that diffusion creep is occurring, a semi-empirical expression for the temperature and pressure dependence of the viscosity is given. The viscosity is found to have a strong minimum near a 100-km depth. The effect of a strong temperature-dependent viscosity on the ascending, diverging flow near ocean ridges is studied. The topography of ocean ridges is attributed to the hydrostatic head required for the horizontal flow. The excess temperature associated with the ascending flow is found to be of the order of 50°K.

D. L. Turcotte

Papers

Convection in a mantle with variable physical properties