Numerical calculations have been carried out to study the effect of the Rayleigh number, viscosity distribution and internal heating on the areal density of upwellings in three-dimensional Cartesian models of mantle convection. Model domain with large aspect ratio (6x6x1) has been applied to ensure a robust ‘upwelling statistics’ and to minimize the distorting effect of the sides of the box. Model parameters were chosen to simulate physical processes occurring in the Earth’s mantle according to available computational resources. The Rayleigh number varied in the domain of Ra=105-5.107, the viscosity increased exponentially with depth by a factor of g=1, 10, 30 or 100, additionally a mantle-like depth-dependent viscosity profile has been studied. The value of internal heating corresponds to the chondritic abundance of radioactive elements, nondimensional internal heating was H=10. Horizontal boundaries were isothermal and stress-free, vertical boundaries had mirror symmetry.
It was found that by increasing the Rayleigh number the flow velocity becomes faster (v~Ra2/3), convective cooling intensifies, thus the surface heat flow increases (Nu~Ra1/3), the characteristic length of the horizontal and vertical boundary layers decreases (d~Ra1/3). Obtained exponents are in accordance with results from scaling analysis [Solomatov 1995]. By increasing the viscosity contrast between the bottom and the top of the box the flow slows down especially in the lower regions resulting in the decrease of the heat transport and causing the strengthening (in thermal sense) of the bottom boundary layer. Therefore, the symmetry of the convective pattern deforms, upwellings are represented as axisymmetric ascending currents (plumes), while downwellings have slablike feature.
On the basis of the numerical investigation of the areal density of upwellings the number of plumes increases with growing Rayleigh number [Parmentier and Sotin 2000], but decreases with the increase of the viscosity contrast. It has been established, that the number of upwellings is related to the thickness of the bottom boundary layer as its inverse proportion, N~d1-1. It is supported by both numerical models and scale analysis. In the presence of internal heating the form of the convection modifies substantially. The average temperature in the model domain increases leading to the strengthening of the upper and the weakening of the bottom thermal boundary layer. As a consequence, downwellings will appear as cold plumes, upwellings as passively ascending diffuse, hot zones. However, by increasing the Rayleigh number the model domain cools, temperature difference across the bottom boundary layer increases. Increasing viscosity contrast decreases more the average temperature of the box making stronger the bottom thermal boundary layer. The high Rayleigh number and the viscosity increasing with depth results in the main forms of convective motion as upwelling plumes and downwelling slabs. Since the internal heating weakens the bottom boundary layer, thus increases the number of plumes. On the basis of flow parameters calculated from numerical models (average surface heat flux, velocity of the lithosphere, thickness of lithosphere and D’’ zone, diameter of plumes) mantle convection occurs rather in one- than two-layered form. The type of the convective flow system (one- or two-layered) can be inferred from the relation between the areal density of upwellings and surface hotspots, as well. If the convection occurs in one-layered form, the density of plumes beneath hotspots will be about 2-3 (using the depth of the mantle as a unit) [Steinberger 2000]. However, if the flow system were two-layered separated by the phase boundary at the depth of 660 km, it would yield the density of upwellings of 0.04-0.06 in the upper mantle (using the depth of the upper mantle as a unit). Whereas the areal density of plumes obtained from our realest numerical models lies in the domain of 0.8-1, which is between the two values, a coexisting one- and two-layered flow regime is supported. This form of the mantle convection is not a new hypothesis, such a flow system has been suggested by both geochemical investigations of extrusive basalts and numerical models including mineral phase transition at 660 km [Cserepes and Yuen 1997].
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Parmentier, E. M., C. Sotin, Three-dimensional numerical experiments on thermal convection in a very viscous fluid: Implications for the dynamics of a thermal boundary layer at high Rayleigh number, Phys. Fluids, 12/3, 2000, 609–617.
Solomatov, V. S., Scaling of temperature- and stress dependent viscosity convection, Phys. Fluids, 7/2, 1995, 266–274.
Steinberger, B., Plumes in a convecting mantle: Models and observations for individual hotspots, J. Geophys. Res., 105, 2000, 11127–11152.