arXiv subjects

P. Garaud

Publications and source records attributed to P. Garaud.

At least 19 recordsLinked to original sources

arXiv2025

Towards a self-consistent hydrodynamical model of the solar tachocline

The solar tachocline is an internal boundary layer in the Sun located between the differentially-rotating convection zone and the uniformly-rotating radiative interior beneath. Spiegel and Zahn (1992) proposed the first hydrodynamical model, which here we call SZ92, arguing that the tachocline is essentially in a steady state of thermal-wind balance, angular-momentum balance, and thermal equilibrium. Angular momentum transport in their model is assumed to be dominated by strongly anisotropic turbulence, primarily horizontal owing to the strong stable stratification of the radiative interior. By contrast, the heat transport is assumed to be dominated by a predominantly vertical diffusive heat flux owing to the thinness of the tachocline. In this paper, we demonstrate that these assumptions are not consistent with the new model of stratified turbulence recently proposed by Chini et al. (2022) and Shah et al. (2024), which has been numerically validated by Garaud et al. (2024). We then propose a simple self-consistent alternative to the SZ92 model, namely, a scenario wherein angular momentum and heat transport are both dominated by horizontal turbulent diffusion. The thickness of the tachocline in the new model scales as $\Omega_\odot / N_m$, where $\Omega_\odot$ is the mean angular velocity of the Sun, and $N_m$ the buoyancy frequency in the tachocline region. We discuss other properties of the model, and show that it has several desirable features, but does not resolve some of the other well-known problems of the SZ92 model.

astro-ph.SR↗

arXiv2024

Constraints on the long-term existence of dilute cores in giant planets

Ring seismology has recently revealed the presence of internal gravity waves inside Saturn that extend up to 60% of Saturn's radius starting from the center, in what is recognized today as Saturn's stably-stratified dilute core. Similarly, gravity measurements on Jupiter suggest the existence of a dilute core of still poorly constrained radial extent. These cores are likely in a double-diffusive regime, which prompt the question of their long-term stability. Indeed, previous DNS (Direct Numerical Simulations) studies in triply-periodic domains have shown that, in some regimes, double-diffusive convection tends to spontaneously form shallow convective layers, which coarsen until the region becomes fully convective. In this letter, we study the conditions for layering in double-diffusive convection using different boundary conditions, in which temperature and composition fluxes are fixed at the domain boundaries. We run a suite of DNS varying microscopic diffusivities of the fluid and the strength of the initial stratification. We find that convective layers still form as a result of the previously discovered gamma-instability which takes place whenever the local stratification drops below a critical threshold that only depends on the fluid diffusivities. We also find that the layers grow once formed, eventually occupying the entire domain. Our work thus recovers the results of previous studies, despite the new boundary conditions, suggesting that this behavior is universal. The existence of Saturn's stably-stratified core, today, therefore suggests that this threshold has never been reached, which places a new constraint on scenarios for the planet's formation and evolution.

astro-ph.EP↗

arXiv2020

The dynamics of stratified horizontal shear flows at low Péclet number

We consider the dynamics of a vertically stratified, horizontally-forced Kolmogorov flow. Motivated by astrophysical systems where the Prandtl number is often asymptotically small, our focus is the little-studied limit of high Reynolds number but low Péclet number (which is defined to be the product of the Reynolds number and the Prandtl number). Through a linear stability analysis, we demonstrate that the stability of two-dimensional modes to infinitesimal perturbations is independent of the stratification, whilst three-dimensional modes are always unstable in the limit of strong stratification and strong thermal diffusion. The subsequent nonlinear evolution and transition to turbulence is studied numerically using direct numerical simulations. For sufficiently large Reynolds numbers, four distinct dynamical regimes naturally emerge, depending upon the strength of the background stratification. By considering dominant balances in the governing equations, we derive scaling laws for each regime which explain the numerical data.

physics.flu-dyn↗

arXiv2020

Horizontal shear instabilities at low Prandtl number

Turbulent mixing in the radiative regions of stars is usually either ignored or crudely accounted for in most stellar evolution models. However, there is growing theoretical and observational evidence that such mixing is present and can affect various aspects of a star's life. In this work, we present a first attempt at quantifying mixing by horizontal shear instabilities in stars using Direct Numerical Simulations. The shear is driven by a body force, and rapidly becomes unstable. At saturation, we find that several distinct dynamical regimes exist, depending on the relative importance of stratification and thermal diffusion (viscosity can in principle also matter, but is usually negligible in most stellar applications). In each of the regimes identified, we put forward a certain number of theoretically motivated scaling laws for the turbulent vertical eddy scale, the typical turbulent diffusion coefficient, and the typical amplitude of temperature fluctuations (among other quantities). Based on our findings, we predict that the majority of stars should fall into one of two categories: high Péclet number stratified turbulence, and low Péclet number stratified turbulence. The latter is presented in detail in a related paper by Cope et al. (2020), while the former is discussed here. Applying our results to the best-known stellar shear layer, namely the solar tachocline, we find that it should lie in the high Péclet number stratified turbulence regime, and predict a substantial amount of vertical mixing for temperature, momentum and composition. Taken as is, the new turbulence model predictions are incompatible with the Spiegel & Zahn (1992) model of the solar tachocline. However, we also show that rotation and magnetic fields are likely to affect the turbulence, and need to be taken into account in future studies.

astro-ph.SR↗

arXiv2019

The interaction between shear and fingering (thermohaline) convection

Fingering convection is a turbulent mixing process that can occur in stellar radiative regions whenever the mean molecular weight increases with radius. In some cases, it can have a significant observable impact on stellar structure and evolution. The efficiency of mixing by fingering convection as a standalone process has been studied by Brown et al. (2013), but other processes such as rotation, magnetic fields and shear can affect it. In this paper, we present a first study of the effect of shear on fingering (thermohaline) convection in astrophysics. Using Direct Numerical Simulations we find that a moderate amount of shear (that is not intrinsically shear-unstable) always decreases the mixing efficiency of fingering convection, as a result of the tilt it imparts to the fingering structures. We propose a simple analytical extension of the Brown et al. (2013) model in the presence of shear that satisfactorily explains the numerically-derived turbulent compositional mixing coefficient for moderate shearing rates, and can trivially be implemented in stellar evolution codes. We also measure from the numerical simulations a turbulent viscosity, and find that the latter is strongly tied to the turbulent compositional mixing coefficient. Observational implications and caveats of the model are discussed.

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arXiv2018

Turbulent transport by diffusive stratified shear flows: from local to global models. Part II: Limitations of local models

This paper continues the systematic investigation of diffusive shear instabilities initiated in Part I of this series. In this work, we primarily focus on quantifying the impact of non-local mixing, which is not taken into account in Zahn's mixing model \citep{Zahn92}. We present the results of direct numerical simulations in a new model setup designed to contain coexisting laminar and turbulent shear layers. As in Part I, we use the Low Péclet Number approximation of \citet{Lign1999} to model the evolution of the perturbations. Our main findings are twofold. First, turbulence is not necessarily generated whenever Zahn's nonlinear criterion \citep{Zahn1974} $J{\rm Pr} < (J{\rm Pr})_c$ is satisfied, where $J=N^2/S^2$ is the local gradient Richardson number, ${\rm Pr} = ν/ κ_T$ is the Prandtl number, and $(J{\rm Pr})_c \simeq 0.007$. We have demonstrated that the presence or absence of turbulent mixing in this limit hysteretically depends on the history of the shear layer. Second, Zahn's nonlinear instability criterion only approximately locates the edge of the turbulent layer, and mixing beyond the region where $J{\rm Pr} < (J{\rm Pr})_c$ can also take place in a manner analogous to convective overshoot. We found that the turbulent kinetic energy decays roughly exponentially beyond the edge of the shear-unstable region, on a lengthscale $δ$ that is directly proportional to the scale of the turbulent eddies, which are themselves of the order of the Zahn scale (see Part I). Our results suggest that mixing by diffusive shear instabilities should be modeled with more care than is currently standard in stellar evolution codes.

astro-ph.SR↗

arXiv2018

The effect of rotation on fingering convection in stellar interiors

We study the effects of rotation on the growth and saturation of the double-diffusive fingering (thermohaline) instability at low Prandtl number. Using direct numerical simulations, we estimate the compositional transport rates as a function of the relevant non-dimensional parameters - the Rossby number, inversely proportional to the rotation rate, and the density ratio which measures the relative thermal and compositional stratifications. Within our explored range of parameters, we generally find rotation to have little effect on vertical transport. However, we also present one exceptional case where a cyclonic large scale vortex (LSV) is observed at low density ratio and fairly low Rossby number. The LSV leads to significant enhancement in the fingering transport rates by concentrating compositionally dense downflows at its core. We argue that the formation of such LSVs could be relevant to solving the missing mixing problem in RGB stars.

astro-ph.SR↗

arXiv2017

Double-diffusive erosion of the core of Jupiter

We present Direct Numerical Simulations of the transport of heat and heavy elements across a double-diffusive interface or a double-diffusive staircase, in conditions that are close to those one may expect to find near the boundary between the heavy-element rich core and the hydrogen-helium envelope of giant planets such as Jupiter. We find that the non-dimensional ratio of the buoyancy flux associated with heavy element transport to the buoyancy flux associated with heat transport lies roughly between 0.5 and 1, which is much larger than previous estimates derived by analogy with geophysical double-diffusive convection. Using these results in combination with a core-erosion model proposed by Guillot et al. (2004), we find that the entire core of Jupiter would be eroded within less than 1Myr assuming that the core-envelope boundary is composed of a single interface. We also propose an alternative model that is more appropriate in the presence of a well-established double-diffusive staircase, and find that in this limit a large fraction of the core could be preserved. These findings are interesting in the context of Juno's recent results, but call for further modeling efforts to better understand the process of core erosion from first principles.

astro-ph.EP↗

arXiv2016

Turbulent transport by diffusive stratified shear flows: from local to global models. Part I: Numerical simulations of a stratified plane Couette flow

Shear-induced turbulence could play a significant role in mixing momentum and chemical species in stellar radiation zones, as discussed by Zahn (1974). In this paper we analyze the results of direct numerical simulations of stratified plane Couette flows, in the limit of rapid thermal diffusion, to measure the turbulent diffusivity and turbulent viscosity as a function of the local shear and the local stratification. We find that the stability criterion proposed by Zahn (1974), namely that the product of the gradient Richardson number and the Prandtl number must be smaller than a critical values $(J\Pr)_c$ for instability, adequately accounts for the transition to turbulence in the flow, with $(J\Pr)_c \simeq 0.007$. This result recovers and confirms the prior findings of Prat et al. (2016). Zahn's model for the turbulent diffusivity and viscosity (Zahn 1992), namely that the mixing coefficient should be proportional to the ratio of the thermal diffusivity to the gradient Richardson number, does not satisfactorily match our numerical data when applied as is. It fails (as expected) in the limit of large stratification where the Richardson number exceeds the aforementioned threshold for instability, but it also fails in the limit of low stratification where the turbulent eddy scale becomes limited by the computational domain size. We propose a revised model for turbulent mixing by diffusive stratified shear instabilities, that now properly accounts for both limits, fits our data satisfactorily, and recovers Zahn's 1992 model in the limit of large Reynolds numbers.

astro-ph.SR↗

arXiv2016

Layer Formation in Sedimentary Fingering Convection

When particles settle through a stable temperature or salinity gradient they can drive an instability known as sedimentary fingering convection. This phenomenon is thought to occur beneath sediment-rich river plumes in lakes and oceans, in the context of marine snow where decaying organic materials serve as the suspended particles, or in the atmosphere in the presence of aerosols or volcanic ash. Laboratory experiments of Houk and Green (1973) and Green (1987) have shown sedimentary fingering convection to be similar to the more commonly known thermohaline fingering convection in many ways. Here, we study the phenomenon using 3D direct numerical simulations. We find evidence for layer formation in sedimentary fingering convection in regions of parameter space where it does not occur for non-sedimentary systems. This is due to two complementary effects. Sedimentation affects the turbulent fluxes and broadens the region of parameter space unstable to the $γ$-instability (Radko 2003) to include systems at larger density ratios. It also gives rise to a new layering instability that exists in $γ-$stable regimes. The former is likely quite ubiquitous in geophysical systems for sufficiently large settling velocities, while the latter probably grows too slowly to be relevant, at least in the context of sediments in water.

physics.flu-dyn↗

arXiv2015

Excitation of gravity waves by fingering convection, and the formation of compositional staircases in stellar interiors

Fingering convection (or thermohaline convection) is a weak yet important kind of mixing that occurs in stably-stratified stellar radiation zones in the presence of an inverse mean-molecular-weight gradient. Brown et al. (2013) recently proposed a new model for mixing by fingering convection, which contains no free parameter, and was found to fit the results of direct numerical simulations in almost all cases. Notably, however, they found that mixing was substantially enhanced above their predicted values in the few cases where large-scale gravity waves, followed by thermo-compositional layering, grew spontaneously from the fingering convection. This effect is well-known in the oceanographic context, and is attributed to the excitation of the so-called "collective instability". In this work, we build on the results of Brown et al. (2013) and of Traxler et al. (2011b) to determine the conditions under which the collective instability may be expected. We find that it is only relevant in stellar regions which have a relatively large Prandtl number (the ratio of the kinematic viscosity to the thermal diffusivity), $O(10^{-3})$ or larger. This implies that the collective instability cannot occur in main sequence stars, where the Prandtl number is always much smaller than this (except in the outer layers of surface convection zones where fingering is irrelevant anyway). It could in principle be excited in regions of high electron degeneracy, during He core flash, or in the interiors of white dwarfs. We discuss the implications of our findings for these objects, both from a theoretical and from an observational point of view.

astro-ph.SR↗

arXiv2010

Dynamics of fingering convection I: Small-scale fluxes and large-scale instabilities

Double-diffusive instabilities are often invoked to explain enhanced transport in stably-stratified fluids. The most-studied natural manifestation of this process, fingering convection, commonly occurs in the ocean's thermocline and typically increases diapycnal mixing by two orders of magnitude over molecular diffusion. Fingering convection is also often associated with structures on much larger scales, such as thermohaline intrusions, gravity waves and thermohaline staircases. In this paper, we present an exhaustive study of the phenomenon from small to large scales. We perform the first three-dimensional simulations of the process at realistic values of the heat and salt diffusivities and provide accurate estimates of the induced turbulent transport. Our results are consistent with oceanic field measurements of diapycnal mixing in fingering regions. We then develop a generalized mean-field theory to study the stability of fingering systems to large-scale perturbations, using our calculated turbulent fluxes to parameterize small-scale transport. The theory recovers the intrusive instability, the collective instability, and the gamma-instability as limiting cases. We find that the fastest-growing large-scale mode depends sensitively on the ratio of the background gradients of temperature and salinity (the density ratio). While only intrusive modes exist at high density ratios, the collective and gamma-instabilities dominate the system at the low density ratios where staircases are typically observed. We conclude by discussing our findings in the context of staircase formation theory.

physics.flu-dyn↗

arXiv2010

Dynamics of fingering convection II: The formation of thermohaline staircases

Regions of the ocean's thermocline unstable to salt fingering are often observed to host thermohaline staircases, stacks of deep well-mixed convective layers separated by thin stably-stratified interfaces. Decades after their discovery, however, their origin remains controversial. In this paper we use 3D direct numerical simulations to shed light on the problem. We study the evolution of an analogous double-diffusive system, starting from an initial statistically homogeneous fingering state and find that it spontaneously transforms into a layered state. By analysing our results in the light of the mean-field theory developed in Paper I, a clear picture of the sequence of events resulting in the staircase formation emerges. A collective instability of homogeneous fingering convection first excites a field of gravity waves, with a well-defined vertical wavelength. However, the waves saturate early through regular but localized breaking events, and are not directly responsible for the formation of the staircase. Meanwhile, slower-growing, horizontally invariant but vertically quasi-periodic gamma-modes are also excited and grow according to the gamma-instability mechanism. Our results suggest that the nonlinear interaction between these various mean-field modes of instability leads to the selection of one particular gamma-mode as the staircase progenitor. Upon reaching a critical amplitude, this progenitor overturns into a fully-formed staircase. We conclude by extending the results of our simulations to real oceanic parameter values, and find that the progenitor gamma-mode is expected to grow on a timescale of a few hours, and leads to the formation of a thermohaline staircase in about one day with an initial spacing of the order of one to two metres.

physics.flu-dyn↗

arXiv2009

On the Penetration of Meridional Circulation below the Solar Convection Zone II: Models with Convection Zone, the Taylor-Proudman constraint and Applications to Other Stars

The solar convection zone exhibits a strong level of differential rotation, whereby the rotation period of the polar regions is about 25-30% longer than the equatorial regions. The Coriolis force associated with these zonal flows perpetually "pumps" the convection zone fluid, and maintains a quasi-steady circulation, poleward near the surface. What is the influence of this meridional circulation on the underlying radiative zone, and in particular, does it provide a significant source of mixing between the two regions? In Paper I, we began to study this question by assuming a fixed meridional flow pattern in the convection zone and calculating its penetration depth into the radiative zone. We found that the amount of mixing caused depends very sensitively on the assumed flow structure near the radiative--convective interface. We continue this study here by including a simple model for the convection zone "pump", and calculating in a self-consistent manner the meridional flows generated in the whole Sun. We find that the global circulation timescale depends in a crucial way on two factors: the overall stratification of the radiative zone as measured by the Rossby number times the square root of the Prandtl number, and, for weakly stratified systems, the presence or absence of stresses within the radiative zone capable of breaking the Taylor-Proudman constraint. We conclude by discussing the consequences of our findings for the solar interior and argue that a potentially important mechanism for mixing in Main Sequence stars has so far been neglected.

astro-ph.SR↗

arXiv2008

The rotation rate of the solar radiative zone

The rotation rate of the solar radiative zone is an important diagnostic for angular-momentum transport in the tachocline and below. In this paper we study the contribution of viscous and magnetic stresses to the global angular-momentum balance. By considering a simple linearized toy model, we discuss the effects of field geometry and applied boundary conditions on the predicted rotation profile and rotation rate of the radiative interior. We compare these analytical predictions with fully nonlinear simulations of the dynamics of the radiative interior, as well as with observations. We discuss the implications of these results as constraints on models of the solar interior.

astro-ph↗

arXiv2008

Assembling the Building Blocks of Giant Planets around Intermediate Mass Stars

We examine a physical process that leads to the efficient formation of gas giant planets around intermediate mass stars. In the gaseous protoplanetary disks surrounding rapidly-accreting intermediate-mass stars we show that the midplane temperature (heated primarily by turbulent dissipation) can reach > 1000 K out to 1 AU. Thermal ionization of this hot gas couples the disk to the magnetic field, allowing the magneto-rotational instability (MRI) to generate turbulence and transport angular momentum. Further from the central star the ionization fraction decreases, decoupling the disk from the magnetic field and reducing the efficiency of angular momentum transport. As the disk evolves towards a quasi-steady state, a local maximum in the surface density and in the midplane pressure both develop at the inner edge of the MRI-dead zone, trapping inwardly migrating solid bodies. Small particles accumulate and coagulate into planetesimals which grow rapidly until they reach isolation mass. In contrast to the situation around solar type stars, we show that the isolation mass for cores at this critical radius around the more massive stars is large enough to promote the accretion of significant amounts of gas prior to disk depletion. Through this process, we anticipate a prolific production of gas giants at ~1 AU around intermediate-mass stars.

astro-ph↗

arXiv2007

On the penetration of meridional circulation below the solar convection zone

Meridional flows with velocities of a few meters per second are observed in the uppermost regions of the solar convection zone. The amplitude and pattern of the flows deeper in the solar interior, in particular near the top of the radiative region, are of crucial importance to a wide range of solar magnetohydrodynamical processes. In this paper, we provide a systematic study of the penetration of large-scale meridional flows from the convection zone into the radiative zone. In particular, we study the effects of the assumed boundary conditions applied at the convective-radiative interface on the deeper flows. Using simplified analytical models in conjunction with more complete numerical methods, we show that penetration of the convectively-driven meridional flows into the deeper interior is not necessarily limited to a shallow Ekman depth but can penetrate much deeper, depending on how the convective-radiative interface flows are modeled.

astro-ph↗

arXiv2007

Growth and migration of solids in evolving protostellar disks I: Methods and Analytical tests

This series of papers investigates the early stages of planet formation by modeling the evolution of the gas and solid content of protostellar disks from the early T Tauri phase until complete dispersal of the gas. In this first paper, I present a new set of simplified equations modeling the growth and migration of various species of grains in a gaseous protostellar disk evolving as a result of the combined effects of viscous accretion and photo-evaporation from the central star. Using the assumption that the grain size distribution function always maintains a power-law structure approximating the average outcome of the exact coagulation/shattering equation, the model focuses on the calculation of the growth rate of the largest grains only. The coupled evolution equations for the maximum grain size, the surface density of the gas and the surface density of solids are then presented and solved self-consistently using a standard 1+1 dimensional formalism. I show that the global evolution of solids is controlled by a leaky reservoir of small grains at large radii, and propose an empirically derived evolution equation for the total mass of solids, which can be used to estimate the total heavy element retention efficiency in the planet formation paradigm. Consistency with observation of the total mass of solids in the Minimum Solar Nebula augmented with the mass of the Oort cloud sets strong upper limit on the initial grain size distribution, as well as on the turbulent parameter $\alphat$. Detailed comparisons with SED observations are presented in a following paper.

astro-ph↗