Rayleigh-Benard stability and the validity of quasi-Boussinesq or quasi-anelastic liquid approximations

The linear stability threshold of the Rayleigh-Benard configuration is analyzed with compressible effects taken into account. It is assumed that the fluid obeys a Newtonian rheology and Fourier's law of thermal transport with constant, uniform (dynamic) viscosity and thermal conductivity in a uniform gravity field. Top and bottom boundaries are maintained at different constant temperatures and we consider here boundary conditions of zero tangential stress and impermeable walls. Under these conditions, and with the Boussinesq approximation, Rayleigh (1916) first obtained analytically the critical value 27pi^4/4 for a dimensionless parameter, now known as the Rayleigh number, at the onset of convection. This manuscript describes the changes of the critical Rayleigh number due to the compressibility of the fluid, measured by the dimensionless dissipation parameter D and due to a finite temperature difference between the hot and cold boundaries, measured by a dimensionless temperature gradient a. Different equations of state are examined: ideal gas equation, Murnaghan's model and a generic equation of state, which can represent any possible equation of state. We also consider two variations of this stability analysis. In a so-called quasi-Boussinesq model, we consider that density perturbations are solely due to temperature perturbations. In a so-called quasi-anelastic liquid approximation, we consider that entropy perturbations are solely due to temperature perturbations. In addition to the numerical Chebyshev-based stability analysis, an analytical approximation is obtained when temperature fluctuations are written as a combination of only two modes. This analytical expression allows us to show that the superadiabatic critical Rayleigh number quadratic departure in a and D from 27pi^4/4 involves the expansion of density up to the degree three in terms of pressure and temperature.Comment: 42 pages, 30 figures, 4 table

A global shear velocity model of the upper mantle from fundamental and higher Rayleigh mode measurements

International audienceWe present DR2012, a global SV-wave tomographic model of the upper mantle. We use an extension of the automated waveform inversion approach of Debayle (1999) which improves our mapping of the transition zone with extraction of fundamental and higher-mode information. The new approach is fully automated and has been successfully used to match approximately 375,000 Rayleigh waveforms. For each seismogram, we obtain a path average shear velocity and quality factor model, and a set of fundamental and higher-mode dispersion and attenuation curves. We incorporate the resulting set of path average shear velocity models into a tomographic inversion. In the uppermost 200 km of the mantle, SV wave heterogeneities correlate with surface tectonics. The high velocity signature of cratons is slightly shallower (approximate to 200 km) than in other seismic models. Thicker continental roots are not required by our data, but can be produced by imposing a priori a smoother model in the vertical direction. Regions deeper than 200 km show no velocity contrasts larger than +/- 1\% at large scale, except for high velocity slabs within the transition zone. Comparisons with other seismic models show that current surface wave datasets allow to build consistent models up to degrees 40 in the upper 200 km of the mantle. The agreement is poorer in the transition zone and confined to low harmonic degrees (<= 10)

Propagation of tectonic waves

Mountain building depends on the disequilibrium between boundary stresses, either at the base of the deforming lithosphere or its lateral boundaries, and buoyancy stresses arising form lateral density variations within the lithosphere itself. On the basis of the thin viscous sheet approximation, we propose a model which accounts for both crustal and lithospheric thicknesses variations. The deformation is controlled by the sum of the moments of density anomalies (i.e. density anomalies times depth) of compositional and thermal origins. The transport of the compositional moment is obtained from the continuity equation while the transport of the thermal moment is obtained from the heat equation. The resulting set of equations controls the coupled behavior of the crust and lithosphere. It shows that various type of solutions can exist: unstable, stable and propagating. When propagation occurs, the crustal and the lithospheric thickness variations are out of phase. The tectonic waves propagate with velocities around 5 mm yr$^{-1}$ that increase with the crustal thickness and decrease with the lithospheric viscosity. We discuss these solutions and argue that continents may in large part be in a domain of propagating tectonic waves

Remarks on compressible convection in Super-Earths

The radial density of planets increases with depth due to compressibility, leading to impacts on their convective dynamics. To account for these effects, including the presence of a quasi-adiabatic temperature profile and entropy sources due to dissipation, the compressibility is expressed through a dissipation number, $\mathcal{D}$, proportional to the planet's radius and gravity. In Earth's mantle, compressibility effects are moderate, but in large rocky or liquid exoplanets (Super-Earths), the dissipation number can become very large. This paper explores the properties of compressible convection when the dissipation number is significant. We start by selecting a simple Murnaghan equation of state that embodies the fundamental properties of condensed matter at planetary conditions. Next, we analyze the characteristics of adiabatic profiles and demonstrate that the ratio between the bottom and top adiabatic temperatures is relatively small and probably less than 2. We examine the marginal stability of compressible mantles and reveal that they can undergo convection with either positive or negative superadiabatic Rayleigh numbers. Lastly, we delve into simulations of convection performed using the exact equations of mechanics, neglecting inertia (infinite Prandtl number case), and examine their consequences for Super-Earths dynamics.Comment: 31 pages, 15 figure

Empirical 3D basis for the internal density of a planet

International audienceVarious papers have discussed the forward relationships between internal density anomalies of a planet and its external gravity field. The inverse modeling, i.e. finding the internal density anomalies from the external potential is known to be highly non unique. In this research note, we explain how a 3D basis can be built to represent the internal density variations which includes a subset that explicitly spans the kernel of the forward gravity operator. This representation clarifies the origin of the non-uniqueness of the gravity sources and implies the existence of a natural minimal-norm inverse for the internal density. We illustrate these ideas by comparing a tomographic model of the mantle to the minimal norm density

Synthetic Tomographic Images of Slabs from Mineral Physics

The mantle structures observed by seismic tomography can only be linked with convection models by assuming some relationships between temperature, density and velocity. These relationships are complex and non linear even if the whole mantle has a uniform composition. For example, the density variations are not only related to the depth dependent thermal expansivity and incompressibility, but also to the distribution of the mineralogical phases that are themselves evolving with temperature and pressure. In this paper, we present a stoichiometric iterative method to compute the equilibrium mineralogy of mantle assemblages by Gibbs energy minimization. The numerical code can handle arbitrary elemental composition in the system MgO, FeO, CaO, Al$_2$O$_3$ and SiO$_2$ and reaches the thermodynamic equilibrium by choosing the abundances of 31 minerals belonging to 14 possible phases. The code can deal with complex chemical activities for minerals belonging to solid state solutions. We illustrate our approach by computing the phase diagrams of various compositions with geodynamical interest (pyrolite, harzburgite and oceanic basalt). Our simulations are in reasonable agreement with high pressure and high temperature experiments. We predict that subducted oceanic crust remains significantly denser than normal mantle even near the core mantle boundary. We then provide synthetic tomographic models of slabs. We show that properties computed at thermodynamic equilibrium are significantly different from those computed at fixed mineralogy. We quantify the three potential contributions of the seismic anomalies (intrinsic thermal effect, changes in mineralogy induced by temperature variations, changes in the bulk composition) and show that they are of comparable magnitudes. Although the accuracy of our results is limited by the uncertainties on the thermodynamic parameters and equations of states of each individual mineral, future geodynamical models will need to include these mineralogical aspects to interpret the tomographic results as well as to explain the geochemical observations

Thermo-Mechanical Adjustment after Impacts during Planetary Growth

The thermal evolution of planets during their growth is strongly influenced by impact heating. The temperature increase after a collision is mostly located next to the shock. For Moon to Mars size planets where impact melting is limited, the long term thermo-mechanical readjustment is driven by spreading and cooling of the heated zone. To determine the time and length scales of the adjustment, we developed a numerical model in axisymmetric cylindrical geometry with variable viscosity. We show that if the impactor is larger than a critical size, the spherical heated zone isothermally flattens until its thickness reaches a value for which motionless thermal diffusion becomes more effective. The thickness at the end of advection depends only on the physical properties of the impacted body. The obtained timescales for the adjustment are comparable to the duration of planetary accretion and depend mostly on the physical properties of the impacted body

An iterative study of time independent induction effects in magnetohydrodynamics

International audienceWe introduce a new numerical approach to study magnetic induction in flows of an electrically conducting fluid submitted to an external applied field B-0. In our procedure the induction equation is solved iteratively in successive orders of the magnetic Reynolds number Rm. All electrical quantities such as potential, currents, and fields are computed explicitly with real boundary conditions. We validate our approach on the well known case of the expulsion of magnetic field lines from large scale eddies. We then apply our technique to the study of the induction mechanisms in the von Karman flows generated in the gap between coaxial rotating disks. We demonstrate how the omega and alpha effects develop in this flow, and how they could cooperate to generate a dynamo in this homogeneous geometry. We also discuss induction effects that specifically result from boundary conditions