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  • Dimitar Vlaykov - Mixing at convective boundaries in stars
    As a problem where multi-dimensional hydrodynamic simulations of stellar interiors can provide clear guidelines, convective boundary mixing has received a lot of attention in recent years. In this contribution we review several recent studies to form a more complete picture of stellar convection and overshooting. We will outline what has been understood, and discuss questions that remain open. This discussion will be supported by data from global simulations of stars produced with the fully compressible MUltidimensional Stellar Implicit Code (MUSIC), and simulations of pre-main sequence, main sequence, and red giant branch stars.

  • Friedrich Kupka - The Challenge of Numerical Simulations of Convection
    Numerical 3D hydrodynamical simulations of convection have become a very popular tool in astrophysics, both to provide a better understanding of this process per se and for quantitative predictions in the fields of stellar structure and evolution, asteroseismology, or stellar activity, among others. Parameter calibration of simple, algebraic models with this type of numerical simulations are in high demand especially in asteroseismology and, as a consequence, exoplanet research. In spite of these incentives, this approach requires care. In this talk, a spot-light will be given on important pitfalls of such 3D simulations, particularly from the perspective of numerical mathematics, and on the fact that even the best 3D simulations cannot overcome basic limitations of simple models that intrinsically neglect essential physics of convective energy transport and mixing.

  • Yannick Ricard - Compressible convection in the mantle of exoplanets
    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 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. We explore 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 in 2D Cartesian geometry performed using the exact equations of mechanics, neglecting inertia (infinite Prandtl number case), and examine their consequences for super-earth dynamics.

  • Mickael Bourgoin - Preferential concentration of inertial particles in turbulent flows
    Particle-laden flows are of relevant interest in many industrial and natural systems. When the carrier flow is turbulent, a striking phenomenon occurs, where particles with inertia tend to segregate in clusters, also leading to depleted regions.This mechanism, called preferential concentration, results from the interaction of the particles with the multi-scale and random structure of turbulence. The exact mechanism at play and the full dynamical consequences still remain however to be unveiled. This lecture will be devoted to recent experimental investigations of clustering of small water droplets in homogeneous and isotropic active-grid-generated turbulence. We investigate the effects of Reynolds number (Rλ, quantifying the turbulence intensity) and particles Stokes number (St, quantifying particles inertia) on preferential concentration. Using Voronoï tesselations, we characterise clustering level and cluster properties (geometry, typical dimensions and fractality). We will show that the exact same Voronoï analysis can be applied to investigate clustering properties of specific topological points of the velocity field of single phase homogeneous isotropic turbulence (obtained for instance from direct numerical simulations) in order to explore the relevance of possible clustering mechanisms, such as effective compressibility due to centrifugal effects (heavy particles sampling preferentially low-vorticity regions) and sweep-stick mechanisms (heavy particles preferentially sticking to low-acceleration points). Finally some new questions, such as the possible existence of super-clusters (clusters of clusters) and the role of particle finite size effects on preferential concentration phenomenon will be briefly discussed.

  • Rene Gassmoeller - Non-adiabatic boundary layers and complex phase transitions: The problem with reference states in Earth's convection
    Mantle convection and lithosphere dynamics in the Earth and other planets can be treated as the slow deformation of a highly viscous fluid, and as such can be described using the compressible Navier-Stokes equations. On Earth-sized planets compressibility is not a dominant effect and density deviations from a reference profile are generally less than a few percent. Therefore, most modelling studies simplify the governing equations. Common approximations assume a depth-dependent reference profile (the anelastic liquid approximation), or drop compressibility altogether (the Boussinesq approximation). In most previous studies, the error introduced by these approximations was small compared to poorly constrained material properties and limited numerical accuracy. However, as model parametrizations have become more realistic, and model resolution has improved, the error by using simplified conservation equations may no longer be negligible: while such approximations may be reasonable for iso-chemical models of mantle plumes or subducted plates traversing the whole mantle, they may be unsatisfactory for layered materials undergoing significant heating or cooling in boundary layers or materials experiencing complex, solution-dependent phase transitions. Here, I will discuss other formulations of the continuity equation that include dynamic density variations due to temperature, pressure and composition without a density reference state. I also discuss the use of entropy instead of temperature and its benefits for modelling complex phase transitions. Quantifying the improvement in accuracy relative to existing formulations in a number of benchmark models allows us to evaluate for which practical applications these effects are important. Finally, I will present numerical aspects of the new formulations and will conclude with remaining challenges. The presented formulations are implemented and tested in the freely available community software ASPECT.

  • Stephane Labrosse - Scaling of internally heated compressible convection
    The thermal evolution of terrestrial planets is paced by the efficiency of convective heat transfer in their rocky mantle. Two types of heat sources produce buoyancy in such layers, the temperature difference between the surface and the hot core and the radiogenic heating source that is distributed in the bulk. The scaling of such mixed heating convection is rather well established for incompressible fluids in the Boussinesq approximation and we extend this scaling to the case of fully compressible fluids, using the Murnaghan equation of state developed by Ricard et al (2022). The convection equations are solved in 2D using Dedalus (Burns et al 2020). The system is controlled by 7 dimensionless numbers but we fix 4 of them to typical planetary values and explore the parameter space formed by Ra, H and Di, the Rayleigh number, internal heating rate and dissipation number. Compared to the Boussinesq case where a slightly stably stratified mean temperature gradient develops in the bulk in presence of internal heating, we find that the mean temperature profile is closer to the isentropic reference when compressibility is considered. Defining super-isentropic Rayleigh and Nusselt numbers allows us to recover classical boundary layer-type scaling laws.

  • Thierry Alboussiere - Perspectives in compressible convection experiments
    Compressible effects in planets and stars are essentially due to the large length-scales of these objects. Reproducing them in a laboratory small-scale experiment is challenging. In the rotor of a centrifuge, one can take advantage of the large apparent gravity field and reach values of the dissipation parameter similar to those of planetary interiors. This can be achieved using gas as convective medium and its equation of state is thus very different from that of condensed matter. Another difference is related to the importance of Coriolis forces. It can be negligible as in Mantle convection or very strong as in core convection. In a rotor centrifuge, its relative importance is bound to be even larger than in core convection, to the point that two-dimensional convection will be realized in most cases. In terms of intensity of convection, as measured by a Rayleigh number, experiments in a rotor centrifuge are capable of reaching much larger values than those accessible to numerical simulation. This gives a specific advantage to the experiments as they are an intermediate step toward the geo and astro-physical objects, where extreme values of Rayleigh numbers can be found.

Double-diffusive convection

  • Remi Bourgeois (with Pascal Tremblin) - Finite volume methods for compressible convection
    We introduce a new class of finite-volume solvers for stratified flows based on a flux-splitting approach. This approach can be easily extended to the MHD system with gravity, and we present first applications of this solver to the study of an extension of the double-diffusive convective theory to MHD leading to triple-diffusive systems. We will discuss the comparison between 2D/3D numerical experiments and theory.

  • Adrien Morison - Fully compressible numerical simulations of double-diffusive convection
    I use the fully compressible finite volume numerical code MUSIC to simulate double-diffusive convection. I focus in particular on the consequences of increasing the compressibility (i.e. departing from Boussinesq-like regimes) in simple setups.

  • Friedrich Kupka (with Florian Zaussinger) - Compressible double-diffusion convection

  • Ulrich Hansen - Double diffusive convection in geological systems
    In most geological systems the density is determined by a least two components, (a) the temperature and (b) the concentration of chemical components. Since the molecular diffusivities of both ingredients are very different ( the chemical diffusivity can be virtually zero) these systems are prone to exhibit double diffusive convection. Systems of investigation range from small scale magma chambers to convective phenomena of planetary scale. Especially the formation of distinct layers in magma chambers has been investigated in the context of double diffusive convection. Layer formation seems a very generic feature in geology and it seems necessary to better understand the mechanisms behind it. For example, the internal structure of the Earth is made up by several layers, while it is unclear how many layers today exist and if all layers are visible to remote sensing techniques. Layering can hardly be explained by gravitational settling. Double-diffusive convection in the diffusive regime, where the fast diffusing component (heat) drives the flow, while the slowly diffusing component (composition) acts as restoring force is believed to a vital mechanism for layer generation. . A relatively high viscosity is typical for geological systems, in general so high that mechanical inertia is virtually negligible and the flow is characterized by an infinite Prandtl number. At the same time, the thermal Rayleigh number is high (Ra > $10^8$), such that thermal inertia plays a dominant role. Another typical feature is the very strong dependence of the viscosity on temperature. Hot material can be orders of magnitude less viscous than cold material.
    We have investigated the double diffusive convection in such a setting, i.e. in a fluid with strongly temperature dependent viscosity (at least 6 orders of magnitude) at infinite Prandtl number. By choosing an initial condition in which a compositional stably stratified fluid overlies a hot reservoir , we mimic the situation in the early Earth after core formation. Differently from earlier experiments we fixed the temperature rather than the heat flux at the lower boundary, resembling a more realistic condition for the boundary between Earth's core and mantle. We ran a series of numerical experiments in 2 and 3D geometries ranging from simple (constant viscosity) to complex rheologies (strongly temperature and stress-dependent viscosity). For a wide range of buoyancy ratios the initial non-layered structure develops into a state with several separately convecting layers. Since temperature is fixed at the bottom, the layer formation is a self organized process, rather than driven externally. In the course of the experiment the number of layers varies and in the run -down cases ( i.e. zero-compositional flux is assumed at all boundaries) the layering finally disappears. Temperature dependent viscosity is found to significantly stabilize the layers. A dynamical model of a planetary mantle , taking into account double diffusive convection in a fluid with complex rheology leads to thermal history model of a planet (especially Earth) which can potentially explain several features , like the existence of chemical reservoirs and also the onset of plate tectonics.
    Further numerical experiments will be presented, carried out to better understand the role of the absolute value of the Rayleigh number on layer formation in the diffusive regime. Also the fingering regime has been investigated , having in mind the thermo-chemical evolution of magma chambers. Here we have concentrated on the effect of compositional and temperature dependent viscosity on the flow.

  • Celine Guervilly - Fingering convection in planetary cores
    We study fingering convection in the context of stably-stratified layers in planetary cores, using hydrodynamical numerical simulations in a rotating spherical shell. In this talk, we will discuss how the size and velocity of the flow structures vary with the stratification and rotation rate. We will also describe the formation of zonal flows in this system.

  • Andreas Tilgner - Experiments on double diffusive finger convection
    Double diffusive systems of interest are usually characterized by two diffusion constants differing by several orders of magnitude. This presents a challenge for numerical simulations. The resulting difficulties are particularly striking in the case of finger convection with staircases when narrow slowly evolving fingers coexist with large scale turbulent convection rolls and all structures need to be resolved. Experimental studies of such systems are therefore essential. This contribution will present experiments which used an electrodeposition cell to sustain a destabilizing concentration difference of copper ions in aqueous solution between the top and bottom boundaries of the cell. The resulting convecting motion is analogous to Rayleigh- Bénard convection at high Prandtl numbers if the cell is kept at spatially uniform temperature. If a stabilizing temperature gradient is imposed across the cell, double diffusive fingers appear even for thermal buoyancy two orders of magnitude smaller than chemical buoyancy. It is found that the fingers obey several simple scaling laws. The control parameters can also be chosen such that fingers and convection rolls coexist in vertically stacked layers, the so called staircases. Staircases are observed even if the density stratification is unstable. The mechanisms governing the transitions from a single convection roll to a single finger layer and a staircase will be discussed.

  • Andrew Cumming - Composition gradients and convection in dense interiors: giant planets and white dwarfs
    I will discuss two situations involving the interaction of composition gradients and convection driven by boundary fluxes: the evolving composition profile of Jupiter's interior and crystallization-driven fingering convection in white dwarfs. Both are motivated by recent observational results - the inferred density profile of Jupiter from the Juno mission, and new results on cooling white dwarfs from Gaia. For both situations, I will discuss the important role of rotation, and likely important effects of compressibility which has yet to be included in simulations. For Jupiter, I will show results from Boussinesq simulations of penetrative convection and layer formation at Prandtl number<1, and discuss the penetration rate and the mechanism for layer formation and how it compares to evolutionary models of Jupiter. These simulations suggest that rotation acts to significantly reduce the efficiency with which heavy elements can be mixed into the convection zone. For white dwarfs, I will discuss the regime of convection driven by chemical separation during crystallization. The large thermal conductivity of the white dwarf generally leads to fingering/thermohaline convection with up-gradient heat transport. I will show the current state of evolutionary models and (Boussinesq) simulations of this kind of convection, and discuss what further work is needed to be able to accurately follow the evolution and to assess whether white dwarf magnetic fields could originate from a crystallization-driven dynamo.

Gravity waves (and associated problems)

  • Kevin Belkacem - Gravity waves and induced transport of angular momentum in radiative interior of low-mass stars
    Transport of angular momentum is a long-standing problem in stellar physics which recently became more acute thanks to the observations of the space-borne mission CoRoT and Kepler. Indeed, the need for an efficient mechanism able to explain the rotation profile of low-mass stars has been emphasized by asteroseimology and waves are among the potential candidates to do so. After reviewing the observational constraints we now have for rotation profiles in low-mass stars, I will focus on how waves are able to transport and redistribute angular momentum in these stars. Particular emphasis will be given on the importance of realistic estimate of wave generation together with the importance of properly considering interactions between meridional circulation and waves.

  • Arthur Le Saux - Studying internal gravity waves in stellar interiors with fully compressible simulations
    To date, the properties of internal gravity waves (IGWs) in stellar interiors remain poorly constrained because of the challenge of observing them. Indeed, in low-mass stars, which possess a convective envelope, they propagate in the inner radiative region and have very small amplitude at the photosphere. In higher mass stars, IGWs are generated at the edge of the convective core and can propagate in the radiative envelope but it remains unclear if they can travel up to the stellar surface.
    Hydrodynamical simulations thus offer a great opportunity to study the properties of IGWs and thus to guide observations. I will present a study of IGWs properties in solar-like and intermediate-mass stars based on 2D stellar structure models preformed with a fully compressible hydrodynamics time implicit code, the MUSIC code. Our results present a good agreement with theoretical predictions concerning waves generation, propagation and damping by radiative effects. In addition, to model realistic IGWs propagating in radiative zone of stars, it is crucial to include a realistic profile of radiative diffusion in hydrodynamical simulations. Especially as it is through radiative damping that waves are able to transport angular momentum and energy.
    Finally, we add a word of caution regarding the interpretation of results from simulations as their numerical set up can have an impact of physical phenomena. Consequently, direct comparison between numerical simulations and observations must be made with caution.

  • Daniel Lecoanet - Generation and Propagation of Convectively Excited Gravity Waves
    Many astrophysical and geophysical systems have regions which are unstable to convective adjacent to stably-stratified regions. In these cases, the convection can excite internal gravity waves. In turn, these waves can transport angular momentum, energy, and chemicals. Here we will present a series of increasingly realistic numerical simulations of convective excitation and subsequent propagation of gravity waves. In all cases, the wave generation matches classic theoretical predictions, and the wave propagation follows from linear theory. This gives robust estimates for the amplitudes of internal gravity waves in astrophysical and geophysical systems, which can then be used to parametrize wave mixing and transport.

  • Guillaume Laibe - Topology for stars and discs pulsations
    Global modes of stellar and disc pulsations are revisited with topological tools inherited from condensed matter physics. We present topological quantities to which the existence and the signature of stable and unstable topological modes are connected, and explain the central role played by compressibility in these systems.

  • Guillaume Roullet - The energetics of mixing in implicit large eddy simulations of oceanic flows
    Implicit large eddy simulations (ILES) are a tool to study flows at very large Reynolds numbers. Compared to explicit LES, ILES do not have an explicit sub-grid closure controlling the key turbulent processes such as mixing and dissipation. In ILES these processes are performed implicitly by the numerics which handles the resolved transport. As a result, these processes seem to emerge from a black box. This is a strong critique to ILES. Here, we present a method to diagnose the local rates of tracer mixing and energy dissipation in ILES. The method is based on extracting the irreversible part of the numerical fluxes. We apply the method to a series of idealized Boussinesq stratified flows, including a convection experiment and a gravity current. These local rates, complemented with the injection rate and the conversion rate allow to monitor the exchanges of energy between the kinetic energy, the available potential energy and the background energy reservoirs. This energy cycle reveals how the flows work, from the energy point of view. The diagnostic also naturally provides the mixing efficiency of the flows, which turns out to crucially depend on the primary source of energy, whether it is kinetic or potential.

  • Joel Sommeria - From internal gravity waves to turbulence and mean flow generation
    An overview of experiments on internal gravity waves and inertio-gravity waves performed on the Coriolis rotating platform at Grenoble will be presented. Regimes of wave turbulence are observed in a closed domain, involving cascade process by wave-wave interactions. Stratified turbulence is obtained at small scales as the outcome of this cascade process. Consequences for vertical mixing and mean flow generation by turbulence will be discussed.

Beyond sphericity: deformation effects due to fast rotation

  • Michel Rieutord - Modelling Rapidly Rotating Stars
    I shall present a few results obtained within the ESTER project on modelling actual stars like Altair, anticipating evolution of massive stars or predicting the spectral energy distribution of fast rotating stars.

  • Thomas Guillet - Directions and prospects for stellar hydrodynamics simulations with MUSIC
    I will present some possible future directions for simulations of stellar hydrodynamics with the MUSIC code, including convection in fast-rotating stars, as well as prospects towards improving our understanding of stellar convection and waves.

  • Thomas Gastine - Numerical models of rotating convection in spherical shells

  • Benjamin Favier - Wall modes in rapidly-rotating convection: an experimental challenge solved by numerical simulations?
    Rapidly-rotating thermal convection is relevant in many planetary cores but remains an asymptotic regime difficult to probe numerically or experimentally. Most existing experimental setups are using a cylindrical container elongated along the rotation axis. Unfortunately, recent measurements show the emergence of strong zonal flows attached to the vertical side wall, which significantly perturb the geophysically-relevant heat flux transported by the bulk flow. I will first discuss their origin and illustrate their surprising robustness, reminiscent of topologically-protected edge states in condensed-matter physics. In a second part, I will show using numerical simulations how to suppress them and potentially achieve the geophysically relevant regime of rotating convection in the lab.

  • Jerome Noir - On the effects of small scale topography in rotating fluids
    Geophysical observations suggest the CMB is not perfectly smooth, nor is the bottom of our oceans, the same is likely to be true in Earth-like planets and icy satellites. How does the small scale topography affect the coupling between the fluid and solid layers in planets remains an open question. Most efforts on this topic focus to oceanic and atmospheric circulation, with a particular emphasis on local and meso-scale dynamics. Numerically very challenging, this topic might be better approach through experiments. I will present some experimental results on the effect of topography in rapidly rotating fluid for quasi-steady flows, Spin-up/Down, and oscillatory inertial waves.

  • David Cébron - Simulations of non-spherical liquid cores of planets
    Current simulations of planetary cores consider spherical shells. Such geometries are indeed well suited for convection, and allow the use of fast spectral codes (based on spherical harmonics). However, certain observations motivate the modelling of non-spherical cores. For instance, the early magnetic fields of the Earth and Moon, as inferred from paleomagnetic data, call for further investigations of the dynamos driven by tides and precession, respectively. Yet, these unusual dynamo mechanisms require large-scale departures from spherical geometries. The opposite small-scale limit is also of interest to explain the dissipative coupling constant used in the Earth’s nutation models. I will present how these various deformations can be modelled for planetary cores by using different complementary methods to account for buoyancy, rotation, and magnetic effects.

  • Michael Le Bars - Turbulence driven by tides and libration
    Rotating turbulence is commonly known for being dominated by geostrophic vortices that are invariant along the rotation axis and undergo an inverse cascade. Yet, another state of inertial wave turbulence is also possible, with weakly non-linearly interacting inertial waves driving fully three-dimensional motions and a downscale energy cascade. This second state could be especially relevant in planetary and stellar non-convecting interiors, where energy is injected through wave excitation via gravitational interactions including tides and libration. The presence of inertial wave turbulence in such systems would then significantly change our current understanding of dissipation, mixing, and dynamo mechanism. In this talk, I will review our experimental and numerical works focusing on the conditions of emergence and on the characteristics of the inertial wave turbulence, and I will discuss the future challenges for planetary and stellar applications.