arXiv subjects

Pascale Garaud

Publications and source records attributed to Pascale Garaud.

At least 37 records · Page 2Linked to original sources

arXiv2018

Convective overshooting and penetration in a Boussinesq spherical shell

We study the dynamics associated with the extension of turbulent convective motions from a convection zone (CZ) into a stable region (RZ) that lies below the latter. For that purpose, we have run a series of three-dimensional direct numerical simulations solving the Navier-Stokes equations under the Boussinesq approximation in a spherical shell geometry. We observe that the overshooting of the turbulent motions into the stably stratified region depends on three different parameters: the relative stability of the RZ, the transition width between the two, and the intensity of the turbulence. In the cases studied, these motions manage to partially alter the thermal stratification and induce thermal mixing, but not so efficiently as to extend the nominal CZ further down into the stable region. We find that the kinetic energy below the convection zone can be modeled by a half-Gaussian profile whose amplitude and width can be predicted a priori for all of our simulations. We examine different dynamical lengthscales related to the depth of the extension of the motions into the RZ, and we find that they all scale remarkably well with a lengthscale that stems from a simple energetic argument. We discuss the implications of our findings for 1D stellar evolution calculations.

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arXiv2018

Turbulent transport by diffusive stratified shear flows: from local to global models. III. A closure model

Being able to account for the missing mixing in stellar radiative zones is a key step toward a better understanding of stellar evolution. Zahn (1974) argued that thermally diffusive shear-induced turbulence might be responsible for some of this mixing. In Part I and Part II of this series of papers we showed that Zahn's (1974, 1992) mixing model applies when the properties of the turbulence are local. But we also discovered limitations of the model when this locality condition fails, in particular near the edge of a turbulent region. In this paper, we propose a second-order closure model for the transport of momentum and chemical species by shear-induced turbulence in strongly stratified, thermally diffusive environments (the so-called low Péclet number limit), which builds upon the work of Garaud \& Ogilvie (2005). Comparison against direct numerical simulations (DNSs) shows that the model is able to predict the vertical profiles of the mean flow and of the stress tensor (including the momentum transport) in diffusive shear flows, often with a reasonably good precision, and at least within a factor of order unity in the worst case scenario. The model is sufficiently simple to be implemented in stellar evolution codes, and all the model constants have been calibrated against DNSs. While significant limitations to its use remain (e.g. it can only be used in the low Péclet number, slowly rotating limit), we argue that it is more reliable than most of the astrophysical prescriptions that are used in stellar evolution models today.

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arXiv2017

Weakly non-Boussinesq convection in a gaseous spherical shell

We examine the dynamics associated with weakly compressible convection in a spherical shell by running 3D direct numerical simulations using the Boussinesq formalism [1]. Motivated by problems in astrophysics, we assume the existence of a finite adiabatic temperature gradient $\nabla T_{\rm{ad}}$ and use mixed boundary conditions for the temperature with fixed flux at the inner boundary and fixed temperature at the outer boundary. This setup is intrinsically more asymmetric than the more standard case of Rayleigh-Bénard convection in liquids between parallel plates with fixed temperature boundary conditions. Conditions where there is substantial asymmetry can cause a dramatic change in the nature of convection and we demonstrate that this is the case here. The flows can become pressure- rather than buoyancy- dominated leading to anomalous heat transport by upflows. Counter-intuitively, the background temperature gradient $\nabla\bar{T}$ can develop a subadiabatic layer (where $\boldsymbol{g}\cdot\nabla\bar{T}<\boldsymbol{g}\cdot\nabla T_{\rm{ad}}$, where $\boldsymbol{g}$ is gravity) although convection remains vigorous at every point across the shell. This indicates a high degree of non-locality.

physics.flu-dyn↗

arXiv2016

The effect of rotation on oscillatory double-diffusive convection (semiconvection)

Oscillatory double-diffusive convection (ODDC, more traditionally called semiconvection) is a form of linear double-diffusive instability that occurs in fluids that are unstably stratified in temperature (Schwarzschild unstable), but stably stratified in chemical composition (Ledoux stable). This scenario is thought to be quite common in the interiors of stars and giant planets, and understanding the transport of heat and chemical species by ODDC is of great importance to stellar and planetary evolution models. Fluids unstable to ODDC have a tendency to form convective thermo-compositional layers which significantly enhance the fluxes of temperature and chemical composition compared with microscopic diffusion. Although a number of recent studies have focused on studying properties of both layered and non-layered ODDC, few have addressed how additional physical processes such as global rotation affect its dynamics. In this work we study first how rotation affects the linear stability properties of rotating ODDC. Using direct numerical simulations we then analyze the effect of rotation on properties of layered and non-layered ODDC, and study how the angle of the rotation axis with respect to the direction of gravity affects layering. We find that rotating systems can be broadly grouped into two categories, based on the strength of rotation. Qualitative behavior in the more weakly rotating group is similar to non-rotating ODDC, but strongly rotating systems become dominated by vortices that are invariant in the direction of the rotation vector and strongly influence transport. We find that whenever layers form, rotation always acts to reduce thermal and compositional transport.

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arXiv2015

Turbulent transport in a strongly stratified forced shear layer with thermal diffusion

This work presents numerical results on the transport of heat and chemical species by shear-induced turbulence in strongly stratified but thermally diffusive environments. The shear instabilities driven in this regime are sometimes called "secular" shear instabilities, and can take place even when the gradient Richardson number of the flow (the square of the ratio of the buoyancy frequency to the shearing rate) is large, provided the Péclet number (the ratio of the thermal diffusion timescale to the turnover timescale of the turbulent eddies) is small. We have identified a set of simple criteria to determine whether these instabilities can take place or not. Generally speaking, we find that they may be relevant whenever the thermal diffusivity of the fluid is very large (typically larger than $10^{14}$cm$^2$/s), which is the case in the outer layers of high-mass stars ($M\ge 10 M_\odot$) for instance. Using a simple model setup in which the shear is forced by a spatially sinusoidal, constant-amplitude body-force, we have identified several regimes ranging from effectively unstratified to very strongly stratified, each with its own set of dynamical properties. Unless the system is in one of the two extreme regimes (effectively unstratified or completely stable), however, we find that (1) only about 10% of the input power is used towards heat transport, while the remaining 90% is viscously dissipated; (2) that the effective compositional mixing coefficient is well-approximated by the model of Zahn (1992), with $D \simeq 0.02 κ_T /J$ where $κ_T$ is the thermal diffusivity and $J$ is the gradient Richardson number. These results need to be confirmed, however, with simulations in different model setups and at higher effective Reynolds number.

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arXiv2015

2D or not 2D: the effect of dimensionality on the dynamics of fingering convection at low Prandtl number

Fingering convection (otherwise known as thermohaline convection) is an instability that occurs in stellar radiative interiors in the presence of unstable compositional gradients. Numerical simulations have been used in order to estimate the efficiency of mixing induced by this instability. However, fully three-dimensional (3D) computations in the parameter regime appropriate for stellar astrophysics (i.e. low Prandtl number) are prohibitively expensive. This raises the question of whether two-dimensional (2D) simulations could be used instead to achieve the same goals. In this work, we address this issue by comparing the outcome of 2D and 3D simulations of fingering convection at low Prandtl number. We find that 2D simulations are never appropriate. However, we also find that the required 3D computational domain does not have to be very wide: the third dimension need only contain a minimum of two wavelengths of the fastest-growing linearly unstable mode to capture the essentially 3D dynamics of small-scale fingering. Narrow domains, however, should still be used with caution since they could limit the subsequent development of any large-scale dynamics typically associated with fingering convection.

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arXiv2015

The stability of stratified spatially periodic shear flows at low Péclet number

This work addresses the question of the stability of stratified, spatially periodic shear flows at low Péclet number but high Reynolds number. This little-studied limit is motivated by astrophysical systems, where the Prandtl number is often very small. Furthermore, it can be studied using a reduced set of "low-Péclet-number equations" proposed by Lignieres [Astronomy & Astrophysics, 348, 933-939, 1999]. Through a linear stability analysis, we first determine the conditions for instability to infinitesimal perturbations. We formally extend Squire's theorem to the low-Péclet-number equations, which shows that the first unstable mode is always two-dimensional. We then perform an energy stability analysis of the low-Péclet-number equations and prove that for a given value of the Reynolds number, above a critical strength of the stratification, any smooth periodic shear flow is stable to perturbations of arbitrary amplitude. In that parameter regime, the flow can only be laminar and turbulent mixing does not take place. Finding that the conditions for linear and energy stability are different, we thus identify a region in parameter space where finite-amplitude instabilities could exist. Using direct numerical simulations, we indeed find that the system is subject to such finite-amplitude instabilities. We determine numerically how far into the linearly stable region of parameter space turbulence can be sustained.

physics.flu-dyn↗

arXiv2015

A New Model for Mixing By Double-Diffusive Convection (Semi-Convection). III. Thermal and Compositional Transport Through Non-Layered ODDC

Oscillatory double-diffusive convection (ODDC) (also known as semi- convection) refers to a type of double diffusive instability that occurs in regions of planetary and stellar interiors which have a destabilizing thermal stratification and a stabilizing mean molecular weight stratification. In this series of papers, we use an extensive suite of three-dimensional (3D) numerical simulations to quantify the transport of heat and chemical species by ODDC. Rosenblum et al. (2011) first showed that ODDC can either spontaneously form layers, which significantly enhance the transport of heat and chemical species compared to mi- croscopic transport, or remain in a state dominated by large scale gravity waves, in which there is a more modest enhancement of the turbulent transport rates. Subsequent studies in this series have focused on identifying under what condi- tions layers form (Mirouh et al. 2012), and quantifying transport through layered systems (Wood et al. 2013). Here we proceed to characterize transport through systems that are unstable to the ODDC instability, but do not undergo spon- taneous layer formation. We measure the thermal and compositional fluxes in non-layered ODDC from both 2D and 3D numerical simulations and show that 3D simulations are well approximated by similar simulations in a 2D domain. We find that the turbulent mixing rate in this regime is weak and can, to a first level approximation, be neglected. We conclude by summarizing the findings of papers I through III into a single prescription for transport by ODDC.

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arXiv2015

Main Sequence Evolution with Layered Semiconvection

Semiconvection - mixing that occurs in regions that are stable when considering compositional gradients, but unstable when ignoring them - is shown to have the greatest potential impact on main sequence stars with masses in the range 1.2 - 1.7 solar masses. We present the first stellar evolution calculations using a prescription for semiconvection derived from extrapolation of direct numerical simulations of double-diffusive mixing down to stellar parameters. The dominant mode of semiconvection in stars is layered semiconvection, where the layer height is an adjustable parameter analogous to the mixing length in convection. The rate of mixing across the semiconvective region is sensitively dependent on the layer height. We find that there is a critical layer height that separates weak semiconvective mixing (where evolution is well-approximated by using the Ledoux criterion) from strong semiconvective mixing (where evolution is well-approximated by using the Schwarzschild criterion). This critical layer height is much smaller than the minimum layer height expected from simulations so we predict that for realistic layer heights, the evolution is nearly the same as a model ran with the Schwarzschild criterion. We also investigate the effects of compositional gradient smoothing, finding that it causes convective cores to artificially shrink in the absence of additional mixing beyond the convective boundary. Layered semiconvection with realistic layer heights provides enough such mixing that stars will still evolve as if the Schwarzschild criterion is employed. Finally, we discuss the potential of detecting such semiconvection and its implication on convective core sizes in solar-like oscillators.

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arXiv2015

A new model for mixing by double-diffusive convection (semi-convection). II. The transport of heat and composition through layers

Regions of stellar and planetary interiors that are unstable according to the Schwarzschild criterion, but stable according to the Ledoux criterion, are subject to a form of oscillatory double-diffusive (ODD) convection often called "semi-convection". In this series of papers, we use an extensive suite of three-dimensional (3D) numerical simulations to quantify the transport of heat and composition by ODD convection, and ultimately propose a new 1D prescription that can be used in stellar and planetary structure and evolution models. The first paper in this series demonstrated that under certain conditions ODD convection spontaneously transitions from an initially homogeneously turbulent state into a staircase of convective layers, which results in a substantial increase in the transport of heat and composition. Here, we present simulations of ODD convection in this layered regime, we describe the dynamical behavior of the layers, and we derive empirical scaling laws for the transport through layered convection.

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arXiv2014

Double-diffusive mixing in stellar interiors in the presence of horizontal gradients

We have identified an important source of mixing in stellar radiation zones, that would arise whenever two conditions are satisfied: (1) the presence of an inverse vertical compositional gradient, and (2) the presence of density-compensating horizontal gradients of temperature (alternatively, entropy) and composition. The former can be caused naturally by any off-center burning process, by atomic diffusion, or by surface accretion. The latter could be caused by rotation, tides, meridional flows, etc. The linear instability and its nonlinear development have been well-studied in the oceanographic context. It is known to drive the formation of stacks of fingering layers separated by diffusive interfaces, called intrusions. Using 3D numerical simulations of the process in the astrophysically-relevant region of parameter space, we find similar results, and demonstrate that the material transport in the intrusive regime can be highly enhanced compared with pure diffusion, even in systems which would otherwise be stable to fingering (thermohaline) convection.

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arXiv2014

Fingering convection induced by atomic diffusion in stars: 3D numerical computations and applications to stellar models

Iron-rich layers are known to form in the stellar subsurface through a combination of gravitational settling and radiative levitation. Their presence, nature and detailed structure can affect the excitation process of various stellar pulsation modes, and must therefore be modeled carefully in order to better interpret Kepler asteroseismic data. In this paper, we study the interplay between atomic diffusion and fingering convection in A-type stars, and its role in the establishment and evolution of iron accumulation layers. To do so, we use a combination of three-dimensional idealized numerical simulations of fingering convection, and one-dimensional realistic stellar models. Using the three-dimensional simulations, we first validate the mixing prescription for fingering convection recently proposed by Brown et al. (2013), and identify what system parameters (total mass of iron, iron diffusivity, thermal diffusivity, etc.) play a role in the overall evolution of the layer. We then implement the Brown et al. (2013) prescription in the Toulouse-Geneva Evolution code to study the evolution of the iron abundance profile beneath the stellar surface. We find, as first discussed by Théado et al. (2009), that when the concurrent settling of helium is ignored, this accumulation rapidly causes an inversion in the mean molecular weight profile, which then drives fingering convection. The latter mixes iron with the surrounding material very efficiently, and the resulting iron layer is very weak. However, taking helium settling into account partially stabilizes the iron profile against fingering convection, and a large iron overabundance can accumulate. The opacity also increases significantly as a result, and in some cases ultimately triggers dynamical convection.

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arXiv2014

Double-Diffusive Convection

Much progress has recently been made in understanding and quantifying vertical mixing induced by double-diffusive instabilities such as fingering convection (usually called thermohaline convection) and oscillatory double-diffusive convection (a process closely related to semiconvection). This was prompted in parts by advances in supercomputing, which allow us to run Direct Numerical Simulations of these processes at parameter values approaching those relevant in stellar interiors, and in parts by recent theoretical developments in oceanography where such instabilities also occur. In this paper I summarize these recent findings, and propose new mixing parametrizations for both processes that can easily be implemented in stellar evolution codes.

astro-ph.EP↗

arXiv2014

Spin-down dynamics of magnetized solar-type stars

It has long been known that solar-type stars undergo significant spin-down, via magnetic braking, during their Main-Sequence lifetimes. However, magnetic braking only operates on the surface layers; it is not yet completely understood how angular momentum is transported within the star, and how rapidly the spin-down information is communicated to the deep interior. In this work, we use insight from recent progress in understanding internal solar dynamics to model the interior of other solar-type stars. We assume, following Gough and McIntyre (1998), that the bulk of the radiation zone of these stars is held in uniform rotation by the presence of an embedded large-scale primordial field, confined below a stably-stratified, magnetic-free tachocline by large-scale meridional flows downwelling from the convection zone. We derive simple equations to describe the response of this model interior to spin-down of the surface layers, that are identical to the two-zone model of MacGregor and Brenner (1991), with a coupling timescale proportional to the local Eddington-Sweet timescale across the tachocline. This timescale depends both on the rotation rate of the star and on the thickness of the tachocline, and can vary from a few hundred thousand years to a few Gyr, depending on stellar properties. Qualitative predictions of the model appear to be consistent with observations, although depend sensitively on the assumed functional dependence of the tachocline thickness on the stellar rotation rate.

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arXiv2013

Large grains can grow in circumstellar discs

We perform coagulation & fragmentation simulations to understand grain growth in T Tauri & brown dwarf discs. We present a physically-motivated approach using a probability distribution function for the collision velocities and separating the deterministic & stochastic velocities. We find growth to larger sizes compared to other models. Furthermore, if brown dwarf discs are scaled-down versions of T Tauri discs (in terms of stellar & disc mass, and disc radius), growth at the same location with respect to the outer edge occurs to similar sizes in both discs.

astro-ph.EP↗

arXiv2013

Chemical Transport and Spontaneous Layer Formation in Fingering Convection in Astrophysics

A region of a star that is stable to convection according to the Ledoux criterion may nevertheless undergo additional mixing if the mean molecular weight increases with radius. This process is called fingering (thermohaline) convection and may account for some of the unexplained mixing in stars such as those that have been polluted by planetary infall and those burning helium-3. We propose a new model for mixing by fingering convection in the parameter regime relevant for stellar (and planetary) interiors. Our theory is based on physical principles and supported by three-dimensional direct numerical simulations. We also discuss the possibility of formation of thermocompositional staircases in fingering regions, and their role in enhancing mixing. Finally, we provide a simple algorithm to implement this theory in one-dimensional stellar codes, such as KEPLER and MESA.

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arXiv2013

Dynamics of the solar tachocline III: Numerical solutions of the Gough and McIntyre model

We present the first numerical simulations of the solar interior to exhibit a tachocline consistent with the Gough and McIntyre (1998) model. We find nonlinear, axisymmetric, steady-state numerical solutions in which: (1) a large-scale primordial field is confined within the radiation zone by downwelling meridional flows that are gyroscopically pumped in the convection zone (2) the radiation zone is in almost-uniform rotation, with a rotation rate consistent with observations (3) the bulk of the tachocline is magnetic free, in thermal-wind balance and in thermal equilibrium and (4) the interaction between the field and the flows takes place within a very thin magnetic boundary layer, the tachopause, located at the bottom of the tachocline. We show that the thickness of the tachocline scales with the amplitude of the meridional flows exactly as predicted by Gough and McIntyre. We also determine the parameter conditions under which such solutions can be obtained, and provide a simple explanation for the failure of previous numerical attempts at reproducing the Gough and McIntyre model. Finally, we discuss the implications of our findings for future numerical models of the solar interior, and for future observations of the Sun and other stars.

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arXiv2012

From dust to planetesimals: an improved model for collisional growth in protoplanetary disks

Planet formation occurs within the gas and dust rich environments of protoplanetary disks. Observations of these objects show that the growth of primordial sub micron sized particles into larger aggregates occurs at the earliest stages of the disks. However, theoretical models of particle growth that use the Smoluchowski equation to describe collisional coagulation and fragmentation have so far failed to produce large particles while maintaining a significant populations of small grains. This has been generally attributed to the existence of two barriers impeding growth due to bouncing and fragmentation of colliding particles. In this paper, we demonstrate that the importance of these barriers has been artificially inflated through the use of simplified models that do not take into account the stochastic nature of the particle motions within the gas disk. We present a new approach in which the relative velocities between two particles is described by a probability distribution function that models both deterministic motion and stochastic motion. Taking both into account can give quite different results to what has been considered recently in other studies. We demonstrate the vital effect of two "ingredients" for particle growth: the proper implementation of a velocity distribution function that overcomes the bouncing barrier and, in combination with mass transfer in high-mass-ratio collisions, boosts the growth of larger particles beyond the fragmentation barrier. A robust result of our simulations is the emergence of two particle populations (small and large), potentially explaining simultaneously a number of long-standing problems in protoplanetary disks, including planetesimal formation close to the central star, the presence of mm to cm size particles far out in the disk, and the persistence of micron-size grains for millions of years.

astro-ph.EP↗