Alain P. Vincent

Hard turbulent thermal convection and thermal evolution of the mantle

This article summarizes the results of hard turbulent convection obtained in laboratory experiments and numerical simulations. Its applications to mantle convection are illustrated by two-dimensional numerical solutions to (1) Newtonian, (2) non-Newtonian convection and (3) Newtonian convection with multiple phase transitions. In Newtonian mantle convection the transition from soft to hard turbulence is marked by the appearance of disconnected plumes. Spectral analysis of the time series of the Nusselt number reveals the presence of a spectral scaling subrange for hard turbulence but not for soft turbulence. In hard turbulence there is correspondence between the spectra in frequency and wavenumber domains. The slope of the seismic wave spectra measured from seismology suggests that the mantle convection today is strongly time-dependent. The transition to hard-turbulence takes place at much lower Nusselt numbers for non-Newtonian than for Newtonian rheology. For the mantle this would have important ramifications. Non-Newtonian plumes behave quite differently from Newtonian ones in that large curvatures are developed in their trajectories in the hard turbulent regime. Mantle convection with phase transitions tends to become more layered with increasing Rayleigh numbers. The style of mantle convection might have changed from a layered to a more whole mantle type of flow with time. Catastrophic overturns associated with strong gravitational instabilities in the transition zone could be responsible for superplume events.

Viewing seismic velocity anomalies with 3‐D continuous Gaussian wavelets

Seismic velocity anomalies (SVA) have traditionally been viewed as spatial objects. We present a new method for looking at SVA, based on a 3-D continuous Gaussian wavelet transform. Local spectra of the seismic anomalies are calculated with the wavelet transforms. Two proxy quantities based on wavelets are used for viewing SVA. These proxy quantities are the 3-D spatial distributions of (1.) the local maxima of the L2-norm of the seismic anomalies, E-max, and (2.) the associated local horizontal wavenumber k-max. The P1200 tomographical model [Zhou 1996] has been used for this purpose. Geographical distributions of E-max and k-max yield information which are not obvious from direct visual inspection of SVA. Some examples are the depth extent of the tectonic boundaries and the inference of a plume-like object beneath the transition zone under Iceland.

Capabilities of 3‐D wavelet transforms to detect plume‐like structures from seismic tomography

The wavelet transform methods have been applied to viewing 3-D seismic tomography by casting the transformed quantities into two proxy distributions, E-max, the maximum of the magnitude of the local spectra about a local point and the associated local wavenumber, k-max. Using a stochastic background noise, we test the capability of this procedure in picking up the coherent structures of upper-mantle plumes. Plumes with a Gaussian shape and a characteristic width up to 2250 km have been tested for various amounts of the signal-to-noise ratios (SNR). We have found that plumes can be picked out for SNR as low as 0.08 db and that the optimal plume width for detection is around 1500 km. For plume width ranging between 700 km and 2000 km, the SNR can be lower than 1 db. This length-scale falls within the range for plume-detection based on the signal-to-noise levels associated with the current global tomographical models.

Four dynamical regimes for a starting plume model

The growth of two-dimensional plumes was modeled numerically to study the dynamics of plumes with Rayleigh numbers in the range of 104 to 108 and Prandtl numbers in the range of 0.025 to 10 000. In this study we deal with a geometry driven by a heated line source, which is different from the basally heated Rayleigh–Benard convection between two horizontal plates. We found four different regimes for plume growth: a diffusive-viscous regime characterized by both thick thermal and velocity boundary layers; an inviscid-diffusive regime with thin velocity and thick temperature boundary layers; a viscous nondiffusive regime with thick velocity boundary layers and thin thermal ones; and an inviscid nondiffusive regime with both thin velocity and narrow thermal boundary layers. We also studied the dependence of the Nusselt number on height for various Rayleigh and Prandtl numbers. We found that plumes with Prandtl numbers as high as 104 grown at a high Rayleigh number (108) are significantly different from plumes...

Continuous wavelet-like filter for a spherical surface and its application to localized admittance function on Mars

Abstract We have developed a 2D isotropic continuous wavelet-like transform for a spherical surface. The transform is simply defined as the surface convolution between the original field and a kernel, based on the zeroth-order Bessel function with a spherical correction. This spherical correction violates the geometric similarity for the various scales of the kernels, which becomes more apparent at longer wavelengths. We found numerically that this transform is practically equivalent to a Gaussian bandpass filter in the spherical harmonic domain. We have applied this wavelet-like transform on the recently acquired Martian gravity and topography fields. Using a ratio constructed locally from these two fields, we have constructed a map describing the lateral variations of the localized admittance function on Mars.

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.

A level set approach to simulate magnetohydrodynamic-instabilities in aluminum reduction cells

Magnetohydrodynamic instabilities at the metal-bath interface in aluminum reduction cells is an important and not fully understood topic. To simulate the two-fluid three-dimensional unstationary flow subject to a background magnetic field, a level set approach is proposed. It features a formulation in terms of the magnetic vector potential to avoid a numerical growth of the divergence of the magnetic field. The same exact projection scheme (with staggered grids) is used for both the velocity field and the magnetic vector potential. Test simulations show that the overall method behaves well in purely hydrodynamic as well as in fully magnetohydrodynamic regimes, in both cases with a single fluid and with two fluids. We also simulate with our technique the metal pad roll instability and trace the behavior of coupled interracial modes.

A finite volume code for meridional circulation in stars

To understand the driving of both meridional circulation and differential rotation in radiative envelopes of stars, one has to solve for 3D mass, momentum, and energy conservation equations for a compressible gas in a central gravity field. In this study, we propose a novel finite volume technique that uses Cartesian geometry thus reducing greatly the complexity of spherical operators. The boundary conditions are efficiently imposed at the surface of the star using the fictitious points technique. We use the anelastic approximation and the Poisson equation for pressure is solved by the Jacobi method which preserves natural symmetries. We present analytical test cases of the fictitious domain technique, and show our results of asymptotic circulation in a model with little stratification and a large viscosity.

Implementation strategies for real-time particle transport solver

Abstract Many problems in physics and engineering involve the transport of solid particles in a turbulent field. In some cases, it is desirable to study the transport of those particles in “real time”. The prediction of erosion in the rotating part of hydraulic turbines is such a problem. This paper presents a semi-analytic predictor-corrector scheme adapted to the case of a rotating frame of reference. Simplification, related to the interpolation scheme required, is discussed as well as a parallel implementation using MPI on 10Base-T Ethernet interconnected workstations. The 3D solver is coupled with a high performance visualization software. Performance then shows a quasi-linear speedup.

On the dynamics of 3-D single thermal plumes at various Prandtl numbers and Rayleigh numbers

Three-dimensional (3-D) numerical simulations of single turbulent thermal plumes in the Boussinesq approximation are used to understand more deeply the interaction of a plume with itself and its environment. In order to do so, we varied the Rayleigh and Prandtl numbers from Ra ∼ 105 to Ra ∼ 108 and from Pr ∼ 0.025 to Pr ∼ 70. We found that thermal dissipation takes place mostly on the border of the plume. Moreover, the rate of energy dissipation per unit mass ε T has a critical point around Pr ∼ 0.7. The reason is that at Pr greater than ∼0.7, buoyancy dominates inertia and thermal advection dominates wave formation whereas this trend is reversed at Pr less than ∼0.7. We also found that for large enough Prandtl number (Pr ∼ 70), the velocity field is mostly poloidal although this result was known for Rayleigh–Bénard convection (see Schmalzl et al. [On the validity of two-dimensional numerical approaches to time-dependent thermal convection. Europhys. Lett. 2004, 67, 390--396]). On the other hand, at small Prandtl numbers, the plume has a large helicity at large scale and a non-negligible toroidal part. Finally, as observed recently in details in weakly compressible turbulent thermal plume at Pr = 0.7 (see Plourde et al. [Direct numerical simulations of a rapidly expanding thermal plume: structure and entrainment interaction. J. Fluid Mech. 2008, 604, 99--123]), we also noticed a two-time cycle in which there is entrainment of some of the external fluid to the plume, this process being most pronounced at the base of the plume. We explain this as a consequence of calculated Richardson number being unity at Pr = 0.7 when buoyancy balance inertia.