V. S. Solomatov

Scaling of temperature‐ and stress‐dependent viscosity convection

Simple scaling analysis of temperature‐ and stress‐dependent viscosity convection with free‐slip boundaries suggests three convective regimes: the small viscosity contrast regime which is similar to convection in a fluid whose viscosity does not depend on temperature, the transitional regime characterized by self‐controlled dynamics of the cold boundary layer and the asymptotic regime in which the cold boundary becomes stagnant and convection involves only the hottest part of the lid determined by a rheological temperature scale. The first two regimes are usually observed in numerical experiments. The last regime is similar to strongly temperature‐dependent viscosity convection with rigid boundaries studied in laboratory experiments.

NUMERICAL INVESTIGATION OF 2D CONVECTION WITH EXTREMELY LARGE VISCOSITY VARIATIONS

Previous experimental studies of convection in fluids with temperature‐dependent viscosity reached viscosity contrasts of the order of 105. Although this value seems large, it still might not be large enough for understanding convection in the interiors of Earth and other planets whose viscosity is a much stronger function of temperature. The reason is that, according to theory, above 104–105 viscosity contrasts, convection must undergo a major transition—to stagnant lid convection. This is an asymptotic regime in which a stagnant lid is formed on the top of the layer and convection is driven by the intrinsic, rheological, temperature scale, rather than by the entire temperature drop in the layer. A finite element multigrid scheme appropriate for large viscosity variations is employed and convection with up to 1014 viscosity contrasts has been systematically investigated in a 2D square cell with free‐slip boundaries. We reached the asymptotic regime in the limit of large viscosity contrasts and obtained s...

Scaling of time-dependent stagnant lid convection: Application to small-scale convection on Earth and other terrestrial planets

Small-scale convection associated with instabilities at the bottom of the lithospheric plates on the Earth and other terrestrial planets occurs in the stagnant lid regime of temperature-dependent viscosity convection. Systematic numerical simulations of time-dependent, internally heated stagnant lid convection suggest simple scaling relationships for a variety of convective parameters and in a broad range of power law viscosities. Application of these scaling relationships to the Earths oceanic lithosphere shows that for either diffusion or dislocation viscosity of olivine, convective instabilities occur in the lower part of the lithosphere between 85 and 100 km depth (the rheological sublayer). “Wet” olivine satisfies constraints on the heat flux and mantle temperature better than “dry” olivine, supporting the view that the upper mantle of the Earth is wet. This is also consistent with the fact that the rheological sublayer is located below the Gutenberg discontinuity which was proposed to represent a sharp change in water content. The viscosity of asthenosphere is (3–6)×1018 Pa s, consistent with previous estimates. The velocities of cold plumes are relatively high reaching several meters per year in the dislocation creep regime. A low value of the heat flux in old continental cratons suggests that continental lithosphere might be convectively stable unless it is perturbed by processes associated with plate tectonics and hot plumes. The absence of plate tectonics on other terrestrial planets and the low heat transport efficiency of stagnant lid convection can lead to widespread melting during the thermal evolution of the terrestrial planets. If the terrestrial planets are dry, small-scale convection cannot occur at subsolidus temperatures.

Stagnant lid convection on Venus

The effect of strongly temperature-dependent viscosity on convection in the interior of Venus is studied systematically with the help of finite element numerical models. For viscosity contrasts satisfying experimental constraints on the rheology of rocks, Venus is likely to be in the regime of stagnant lid convection. This regime is characterized by the formation of a slowly creeping, very viscous lid on top of the mantle-Venusian lithosphere and is in agreement with the tectonic style observed on Venus. Stagnant lid convection explains large geoid to topography ratios on Venus by the thermal thinning of a thick lithosphere. The thickness of the lithosphere can be as large as 400-550 km for Beta Regio and 200-400 km on average. Geoid and topography data and experimental data on the rheology of rocks provide constraints on the viscosity of the mantle, 10 20 -10 21 Pa s ; the convective stresses in the interior, 0.2-0.5 MPa ; the stresses in the lid, 100-200 MPa ; the velocity in the interior, 0.5-3 cm yr -1 ; and the heat flux beneath the lithosphere, 8-16 mW m -2 . Parameterized convection calculations of thermal history of Venus are difficult to reconcile with a thick present-day lithosphere. However, a sufficiently thick lithosphere can be formed if a convective regime with mobile plates was replaced by stagnant lid convection around 0.5 b.y. ago. One of the possible explanations for the cessation of Venusian plate tectonics is that during the evolution of Venus, stresses in the lid dropped below the yield strength of the lithosphere. This model predicts a drastic drop in the heat flux, thickening of the lithosphere, and suppression of melting and is consistent with the hypothesis of cessation of resurfacing on Venus around 0.5 b.y. ago.

Three regimes of mantle convection with non-Newtonian viscosity and stagnant lid convection on the terrestrial planets

Numerical simulations of convection with strongly temperature-dependent viscosity suggest that non-Newtonian viscosity convection (dislocation creep) passes through three convective regimes similar to those observed for Newtonian viscosity convection (diffusion creep): the small viscosity contrast regime, the transitional regime and the stagnant lid regime. For realistic viscosity contrasts, mantle convection is in the stagnant lid regime characterized by formation of a very viscous, slowly creeping lid on top of an actively convecting mantle. This explains the tectonic style observed on the terrestrial planets and the Moon. On the other hand, this eliminates the possibility that the plates on Earth could be mobile due to non-Newtonian viscosity. The nature of the mobility of lithospheric plates on Earth has yet to be explained.

SUSPENSION IN CONVECTIVE LAYERS AND STYLE OF DIFFERENTIATION OF A TERRESTRIAL MAGMA OCEAN

Recent physical theories for the formation of the Earth suggest that about 4.5 b.y. ago the mantle of the Earth was partially or completely molten. Fractional crystallization of this hypothetical magma ocean would result in a strong chemical stratification of the Earths mantle. Such a scenario is controversial from the geochemical point of view. However, it has been noted that the simple scenario of fractional crystallization could be avoidable in a convective magma ocean if crystals remain suspended. In this paper, the problem of suspension is developed with the help of an energetic approach: convection must do some work against gravitational settling. We distinguish three regimes of convective suspensions. Absolute or complete sedimentation occurs when the energy dissipation due to the settling exceeds the heat loss from the convective layer. This is possible only in large-scale systems like magma oceans and implies that cooling can proceed only together with sedimentation, crystallization, and a decrease in the liquidus temperature at a constant pressure. A regime of partial differentiation occurs when the energy dissipation due to the settling is less than the total heat loss but larger than the power which can be spent by convection on the crystal reentrainment process. The differentiation is not complete, and a competition between the rate of cooling, the rate of sedimentation, and the rate of turbulent diffusion determines the degree of differentiation. The third regime is an absolute suspension which could be sustained for an indefinitely long time. In this case, sedimentation starts only when the crystal fraction reaches the maximum packing value: when the viscosity of the magma rapidly increases. The power which can be spent by convection on reentrainment is equal to eαgd/cp of the total energy supply to the convective layer, where e 15 GPa) or 10^(−3) – 10^(−1) cm during crystallization of shallow layers, the first regime (“fractional crystallization”) is unavoidable. The estimates depend on various poorly constrained parameters and processes, such as heat flux, viscosity, thermodynamical disequilibrium and highly variable viscosity convection. For absolute suspension the crystal size must be at least e^(½) times less, or 10^(−3) – 10^(−1) cm and 10^(−4) – 10^(−2) cm, respectively, if e ∼ 0.01. The partial differentiation occurs in a narrow (one decade) range between these two regimes. The radius of about 1 cm must be considered as an absolute upper bound above which fractional differentiation is guaranteed. These estimates for the critical crystal size are orders of magnitude lower than suggested previously, and thus the problem of crystal sizes becomes a central one for magma oceans. A necessary condition for reentrainment is the existence of local mechanisms. The absence of such mechanisms to reentrain the particles from the bottom would mean that an absolute suspension is impossible even if the energetics allows it. Turbulence is considered as a possible important factor. A simple model of convection predicts a strong turbulence, provided the viscosity is less than 10^9 – 10^(10) P. Rotation reduces this critical viscosity to 10^5 – 10^8 P but this is still sufficiently large and is reached only near the maximum packing crystal fraction. Power law or Bingham rheology of partial melts can exclude any turbulence already at 20 – 30% of crystal fraction. We also show that the energetic criterion for the absolute suspension with e ∼ 1 coincides with the condition that the particle concentration gradient suppresses the turbulence.

Heat transport efficiency for stagnant lid convection with dislocation viscosity: Application to Mars and Venus

Mantle convection on Mars and Venus is likely to occur in the regime known as stagnant lid convection. We perform thermal boundary layer analyses as well as finite element simulations of stagnant lid convection with non-Newtonian viscosity (which is believed to be more appropriate for the lithosphere and upper mantle) and discuss one particular application of the results, the efficiency of heat transport on the terrestrial planets. As in the case of Newtonian viscosity, the efficiency of heat transfer in the stagnant lid regime is extremely low compared to plate tectonics: For example, in the absence of plate tectonics, the mantle temperature on Earth, which is already close to the solidus, would be about 700–1500 K higher for the present-day value of the surface heat flux. For Venus, the critical heat flux which can be removed without widespread melting is only 10–20 m W/m2. For Mars, it is 15–30 m W/m2. Therefore, there are no doubts that in the absence of mobile plates, the mantle temperature would significantly exceed solidus during planetary evolution. It is hypothesized that this could cause one, or a combination, of two possible processes: (1) differentiation of radiogenic isotopes into the crust during early planetary magmatism and (2) initiation of some kind of plate tectonics as a result of plate weakening due to melting.

Nonfractional Crystallization of a Terrestrial Magma Ocean

It has been suggested that evolution of a terrestrial magma ocean does not unavoidably follow a fractional crystallization scenario. Convection is able to preclude differentiation until a sharp viscosity increase occurs near some critical crystal fraction. However, this kind of crystallization and its physical and chemical consequences have not been previously studied. We consider an end-member, called here nonfractional crystallization. We begin with a simple equilibrium thermodynamical model of partial melts which is based on an ideal three-component phase diagram. It allows a self-consistent calculation of physical and chemical parameters in the melting range at all interesting pressures. In particular, adiabats of the convecting magma ocean are calculated. The sharp increase in the viscosity is supposed to occur near the maximum packing crystal fraction. However, almost independently of this value, convection occurs even in the highly viscous quasi-solid part of the magma ocean and it is strong enough to prevent differentiation in deep regions. A kind of compositional convection occurs due to the layered differentiation, although it is weaker than the thermal convection. Only a surface region undergoes an essential differentiation via melt expulsion by compaction. The thickness of this layer depends on the rheology of partial melts, critical crystal fraction, and crystal sizes but in any case the basal pressure hardly can exceed 5 – 10 GPa. Because of lower pressures in the Moon, the thickness of the differentiating layer is large and thus the entire lunar magma ocean could undergo a strong differentiation. Remelting due to the energy released by differentiation is crucial only for much deeper layers (possibly deeper than about 1000 km for the Earth). For the remaining shallow layer (p < 5 – 10 GPa) the predicted increase of the melt fraction is less than 40 % at the surface and decreases to zero at the bottom of the differentiating layer. Thus, the nonfractional crystallization is suggested to be a likely alternative to the fractional crystallization. The crucial and still poorly understood factors are suspension in convective layers, rheology of partial melts, crystal size, and surface conditions. The most pronounced chemical consequence of the nonfractional crystallization is an almost completely preserved undifferentiated lower mantle and possibly a significant undifferentiated part of the upper mantle. At all depths, in the beginning of differentiation not only the first liquidus solid phase but also subsequent phases have been partially crystallized. So, when the differentiation begins, it involves mixtures of phases. It is important for the remaining layer where differentiation is unavoidable: this layer does not have as strong differentiation of minor elements as in the case of fractional crystallization but it will still involve differentiation of major elements. Future geochemical calculations of this multiphase differentiation, considering both major and minor elements, could help to constrain the differentiation further.

Kinetics of crystal growth in a terrestrial magma ocean

The problem of crystal sizes is one of the central problems of differentiation of a terrestrial magma ocean and it has been an arbitrary parameter in previous models. The crystal sizes are controlled by kinetics of nucleation and crystal growth in a convective magma ocean. In contrast with crystallization in magma chambers, volcanic lavas, dikes, and other relatively well studied systems, nucleation and crystallization of solid phases occur due to the adiabatic compression in downward moving magma (adiabatic “cooling”). This problem is solved analytically for an arbitrary crystal growth law, using the following assumptions: convection is not influenced by the kinetics, interface kinetics is the rate controlling mechanism of crystal growth, and the adiabatic cooling is sufficiently slow for the asymptotic solution to be valid. The problems of nucleation and crystal growth at constant heat flux from the system and at constant temperature drop rate are shown to be described with similar equations. This allows comparison with numerical and experimental data available for these cases. A good agreement was found. When, during the cooling, the temperature drops below the temperature of the expected solid phase appearance, the subsequent evolution consists of three basic periods: cooling without any nucleation and crystallization, a short time interval of nucleation and initial crystallization (relaxation to equilibrium), and slow crystallization due to crystal growth controlled by quasi-equilibrium cooling. In contrast to previously discussed problems, nucleation is not as important as the crystal growth rate function and the rate of cooling. The physics of this unusual behavior is that both the characteristic nucleation rate and the time interval during which the nucleation takes place are now controlled by a competition between the cooling and crystallization rates. A probable size range for the magma ocean is found to be 10^(−2) − 1 cm, which is close to the upper bound for the critical crystal size dividing fractional and nonfractional crystallization discussed elsewhere in this issue. Both the volatile content and pressure are important and can influence the estimate by 1–2 orders of magnitude. Different kinds of Ostwald ripening take place in the final stage of the crystal growth. If the surface nucleation is the rate-controlling mechanism of crystal growth at small supercooling, then the Ostwald ripening is negligibly slow. In the case of other mechanisms of crystal growth, the crystal radius can reach the critical value required to start the fractional crystallization. It can happen in the latest stages of the evolution when the crystals do not dissolve completely and the time for the ripening is large.

Can sharp seismic discontinuities be caused by non-equilibrium phase transformations?

A long-posed problem of the seismic discontinuities at the depths of 410 and 660 km in Earths mantle is whether these discontinuities are caused by phase transitions alone or together with chemical layering. Among several tests, the sharpness of the boundary is one of the most crucial. Recent data suggest that the transition region thickness is less than 4 km for both the 410 and the 660 km discontinuities. This may be smaller than predicted by phase equilibria. We suggest that if there is a sufficiently large nucleation barrier for phase transformation and if the transformation front cannot move faster than the convective flow, the transformation in rising or descending material does not take place until the metastable overshoot becomes equal to the nucleation barrier. An avalanche-like transformation following this overshoot occurs in a very narrow region and can be the cause of a sharp seismic discontinuity. In this case, the topography of the phase boundary is also substantially modified.