Andrei V. Malevsky

Various influences on three-dimensional mantle convection with phase transitions

Abstract Three-dimensional numerical calculations of mantle convection with the two major phase transitions have been carried out in a 5 × 5 × 1 configuration to study the effects of increasing vigor of convection, internal heating and extremely negative Clapeyron slope of the spinel-perovskite phase transition. Depth-dependent properties of thermal expansivity, viscosity and thermal conductivity have been incorporated. Three-dimensional solutions for surface Rayleigh number (Ra s ) between 2 × 10 6 and 4 × 10 8 show that there is a distinct transition between Ra s = 4 × 10 7 and 10 8 in which the system changes from single-layered to layered convection with the mass flux decreasing to below 10% at Ra s = 10 8 . Surface heat flux does not decrease with increasing Ra s and with the accompanying decrease in the mass flux at the transition zone. The effects of internal heating are to reduce the wavelengths of the planforms which cause greater degree of layering in the system. Comparison between 2D and 3D results shows that there is a greater mass flux passing through the transition zone in the 2D models for the 5 × 5 × 1 ☐, but for larger aspect ratio (8 × 8 × 1) the 3D flows become more layered than the corresponding 2D solution. Increasing the magnitude of the negative Clapeyron slope by three times the experimental value can bring about a dramatic reduction in the amount of mass flux across the 670 km discontinuity. Results from a 8 × 8 × 1 ☐ show that a greater amount of layering is produced in the larger aspect-ratio configuration because of the shorter wavelengths of the developed planforms. Increasing the degrees of freedom in a 3D system by either greater amounts of convective vigor or arger domains may give rise to a greater tendency for layered convection. Three-dimensional spherical shell models may produce a greater degree of layering than the corresponding axisymmetric models.

Characteristics‐based methods applied to infinite Prandtl number thermal convection in the hard turbulent regime

The use of characteristics‐based methods for the advection‐dominated regimes in thermal convection is investigated. An operator‐splitting method applied to the advection–diffusion equation for the very large Peclet (Pe) number regime is presented. In this approach two partial differential equations representing both the purely hyperbolic and the parabolic components must be solved simultaneously. This method has been compared with (1) the Galerkin approximation, (2) the streamwise upwinding Petrov–Galerkin method, and (3) the characteristics‐based method using the Lagrangian formulation for the time‐derivative operator of the advection–diffusion equation. Solution accuracy of the operator‐splitting method improves with larger Pe, while the accuracy of other methods deteriorates with Pe. For the nonlinear problem of two‐dimensional thermal convection the Lagrangian method is found to be most computationally efficient. With this Lagrangian method, time‐dependent, thermal convection solutions of extremely hi...

Plume structures in the hard-turbulent regime of three-dimensional infinite Prandtl number convection

Numerical simulations of three-dimensional infinite Prandtl number thermal convection with Rayleigh number (Ra) up to 108 are reported. Convection with Ra higher than 107 is characterized by the appearance of disconnected thermal plumes. The smaller plumes are detached by the currents produced by the larger plumes. The low wavenumber portion of a thermal power spectrum near the boundary layer becomes flat at high Ra, while the spectrum measured in the interior shows a positive slope for low wavenumbers. Differences are found in the thermal spectra between 2D and 3D models.

Temperature‐dependent Newtonian and non‐Newtonian convection: Implications for lithospheric processes

Convection studies of temperature-dependent Newtonian and non-Newtonian rheology in aspect-ratio seven boxes have been carried out for volume averaged Rayleigh numbers between O(105) and O(107). Large lateral viscosity variations, O(104) are found for non-Newtonian cases with total temperature-dependent viscosity variations up to 250. Flow structures for both rheologies are distinguished by large-scale circulations with relatively stable descending limbs. Much larger hot thermal anomalies are found near the surface for non-Newtonian rheology. Lithospheric thinning is facilitated by non-Newtonian rheology because of stress-softening and the lubrication of the descending limbs by hot diapirs.

Viscosity and thermal fields associated with strongly chaotic non-Newtonian thermal convection

We have investigated the thermomechanical structure in strongly chaotic non-Newtonian thermal convection for both base-heated and internally-heated systems. Temperature can build up in stagnant regions and a non-Newtonian mantle can tolerate less internal-heating. Viscosity fields of strongly chaotic regime show a granular structure. The horizontal spectra of viscosity fluctuations obey a power-law and yield a fractal dimension of 1.6 to 1.8 for the isoviscosity lines, providing evidence for two-dimensional turbulence. Long wavelength viscosity variations are smoothed out by the turbulent non-Newtonian flows.

The evolution of material surfaces in convection with variable viscosity as monitored by a characteristics-based method

We have studied the evolution of material surfaces in strongly time-dependent convection for both Newtonian and non-Newtonian temperature- and depth-dependent rheologies. A spline-characteristic method has been employed. The method of characteristics is second-order in time and fourth-order in space. Our strategy is to employ a very dense grid for the material surface field with about 50 to 100 times more points than for the temperature field grid. We have detected sharp entrainment of the surrounding material into both the ascending and descending flow structures. We have observed the breakoff process involving the descending plume. Mixing takes place very differently for the two rheologies. With time, small vortical features are developed inside the Newtonian plumes, while unmixed islands still persist in the non-Newtonian flow.

Dynamics of thermal convection with Newtonian temperature-dependent viscosity at high Rayleigh number

Two-dimensional, time-dependent convection with a Newtonian temperature-dependent viscosity has been investigated in wide aspect-ratio boxes. Large-scale circulations have been found in the regime with volumetrically averaged Rayleigh numbers, Rav ranging between O(105) and O(108). Viscosity contrasts up to 1000 have been examined. The horizontally averaged temperature in the interior, 〈T〉12, does not vary much within the above range of Rav for large-aspect-ratio boxes. The horizontal Fourier spectra of the viscosity field show a decrease of magnitude in the viscosity with shorter wavelength. The viscosity spectra decay differently with decreasing wavelength than the corresponding thermal spectra. Lithospheric processes, such as lubrication of subduction and lithospheric thinning, are facilitated by high Rav. The vertical correlation function for the temperature anomalies used recently for comparing convection calculations with tomographic models is found to be strongly time-dependent for Rav ⩽ O(106). The temporal fluctuations of the vertical correlation function decrease for higher Rayleigh numbers. Correlation functions for low Ra are broad and smooth, while those for high Ra, greater than 107, are narrow and fragmented.

Mantle rheology, convection and rotational dynamics

Abstract We have examined theoretically the effects of mantle convection on Earth rotational dynamics for both viscoelastic and viscous mantles. Strategies for numerical computations are proposed. A linear Maxwell viscoelastic rheology accounting for finite deformations associated with mantle convection is considered. For both rheologies the two sets of convection and rotational equations can be partitioned into separate systems with the output from convection being used as input for the rotational equations. The differences in this convection-rotational problem between finite-strain and small-amplitude viscoelastic theories are delineated. An algorithm based on the usage of massively parallel processors is proposed in which all of the different processes in the convection-rotational problem are partitioned and the different timescales can be dealt with together. The coupled systems of convective-rotational equations can greatly be simplified by using the hydrostatic approximation for the rotational readjustment process in a viscous Earth model. This is valid for a young Earth and for non-Newtonian rheology. More contributions to the relative angular momentum can be expected from non-Newtonian rheology. The non-hydrostatic equatorial bulge may also be explained as a consequence of the long-wavelength dynamics associated with the effects of depth-dependent physical properties on mantle convection.

The origin of a characteristic frequency in hard thermal turbulence

New methods are proposed for filtering the time series of heat flux which is useful for detecting the characteristic frequencies in hard turbulent convection. High‐resolution solutions have been obtained for a Rayleigh (Ra) of 108 and infinite Prandtl number in a box with an aspect ratio of 1.8, in which the finest grid consisting of 140×400 bicubic splines was used. Successively higher temporal derivatives and high‐pass spectral filtering of the Nusselt number at this high Ra reveal the existence of bursts. They are closely related to the presence of plumes in the thermal boundary layer in that there is a relative absence of activity in the boundary layer just before the onset of a burst and, on the other hand, there is a period of intense activity shortly before the end of a burst. These bursts are spaced evenly in time, thus yielding a single characteristic frequency, which may be related to a period associated with a pulsation mechanism in hard turbulent convection.

Strongly Chaotic Newtonian and Non-Newtonian Mantle Convection

The topic of hard turbulent thermal convection is discussed with application to the earth sciences. These new ideas, stimulated by laboratory experiments initiated by physicists, may be useful for understanding the dynamics of magma oceans developed in the early stages of terrestrial planets. Results drawn from two-dimensional numerical simulations of base-heated thermal convection for both linear and non-linear rheologies are presented to show by visualization of the temperature, viscosity and vorticity fields in the transition to the hard turbulent regime. The vorticity fields may provide a good diagnostic measure for the transition to hard turbulence. Both the vorticity and viscosity fields in the non-Newtonian hard turbulent regime reveal small-scale motions and the contours of these heterogeneities assume a fractal-like appearance. The threshold Nusselt number, around 20, for the transition to hard turbulence is lower for base-heated non-Newtonian convection than for Newtonian solutions, thus making it possible for the Earth’s upper mantle to be in a hard-turbulent state today.