The phenomenon of thermal and compositional (chemical) convection is very common in nature and therefore of great importance from the point of view of understanding of many fundamental aspects of the environment and universe. A number of books have been written on the topic, such as e.g. the seminal work of Chandrasekhar (1961) on Hydrodynamic and Hydromagnetic Stability, a large portion of which is devoted to the convective instability near its onset or the outstanding book of Getling (1998) where systematization of the knowledge on convection has been continued with a thorough description of the weakly nonlinear stages. Most of the works, however, considered weakly stratified, that is the so-called Boussinesq convection.
It is the aim of this book to continue the process of systematization. It seems important to put the current knowledge on weakly and strongly stratified convection in order and provide a comprehensive description of the marginal, weakly nonlinear and fully developed stages of convective flow in both cases. To that end the book provides a short compendium of knowledge on the linear and weakly nonlinear limits of the Boussinesq convection, as a useful reference for a reader and than proceeds with a review of the theory on fully developed, weakly stratified convection. The entire third chapter is devoted to a detailed derivation and a study of the three aforementioned stages of stratified (anelastic) convection. The description of stratified convection requires extreme care, since many aspects have to be considered simultaneously for full consistency. Detailed and systematic explanations are therefore provided. This book is meant as a textbook for courses on hydrodynamics and convective flows, for the use of lecturers and students, however, it may also be of use for the entire scientific community as a practical reference.
The paper is devoted to the study of the formation of stratification in an incompressible fluid due to convective laminar flows in horizontal layers heated from the side. Medium and intensive modes of stationary laminar thermal, concentrational and thermo-concentrational (in particular thermohaline) convection are considered, in which nonlinear flow features are manifested, which can radically change the flow structure and the characteristics of heat and mass transfer. The solutions of the problems of laminar problems of convection show the features of the formation of layered structures, vert…
Natural convection is usually complicated by additional factors such as rotation, shear, radiative transfer, compressibility and electromagnetic fields (in the case of electro-conductive fluids). It is shown, using results of numerical simulations and measurements in atmospheric boundary layer and solar photosphere that strong stratification can transform turbulence into deterministic chaos with exponential spectral decay of kinetic energy. When the stratification becomes weaker the deterministic chaos is replaced by the distributed chaos with stretched exponential spectral decay controlled by…
A new mechanism of double-component convection is discovered. It emerges in a horizontal layer of Boussinesq fluid as a stable stratification due to flux boundary conditions is added to an unstable gradient specified by fixed boundary values. A large enough perturbation substantially decreases the stable flux gradient but fails to mix the unstable fixed-value gradient. Steady finite-amplitude flows reminiscent of Rayleigh--Benard convection then arise even as the net background stratification is stable, emphasizing the importance of the identified mechanism for double-component fluid systems. …
We consider a vertical cavity composed of two chambers separated by a retractable thermally insulated thin membrane. The upper and lower chambers are filled with an incompressible Boussinesq fluid and maintained at temperatures $T_2$ and $T_1>T_2$, respectively by two separate heaters. Upon retraction of the membrane, the two fluid masses form an unstably stratified configuration with cold and heavy fluid overlying a warmer and lighter fluid and separated by a non-free interface across which there is a jump in the density. The aim of this paper is to determine the threshold conditions for conv…
We study turbulent flows in planar channels with unstable thermal stratification, using direct numerical simulations in a wide range of Reynolds and Rayleigh numbers and reaching flow conditions which are representative of asymptotic developed turbulence. The combined effect of forced and free convection produces a peculiar pattern of quasi--streamwise rollers occupying the full channel thickness with aspect--ratio considerably higher than unity; it has been observed that they have an important redistributing effect on temperature and momentum. The mean values and the variances of the flow var…
The present work focuses on the study of mixed convection of a purely viscous shear-thinning fluid in a horizontal annular eccentric duct. The inner and outer cylinders are heated with constant and uniform heat flux densities. The objective of this work is to study the effect of the variation of eccentricity, rheological behavior of the fluid as well as the thermodependency of the rheological parameters on the reorganization of the flow and thermal stratification caused by the buoyancy forces. At the entrance of the heating zone, the dynamic regime is assumed to be established and the temperat…
We investigate the processes that lead to the generation of mean flows in two-dimensional anelastic convection. The simple model consists of a plane layer that is rotating about an axis inclined to gravity. The results are two-fold: firstly we numerically investigate the onset of convection in three-dimensions, paying particular attention to the role of stratification and highlight a curious symmetry. Secondly, we investigate the mechanisms that drive both zonal and meridional flows in two dimensions. We find that, in general, non-trivial Reynolds stresses can lead to systematic flows and, usi…
Overshooting of turbulent motions from convective regions into adjacent stably stratified zones plays a significant role in stellar interior dynamics as this process may lead to mixing of chemical species, and contribute to the transport of angular momentum and magnetic fields. We present a series of fully non-linear, three-dimensional (3D) anelastic simulations of overshooting convection in a spherical shell which are focused on the dependence of the overshooting dynamics on the density stratification and the rotation, both key ingredients in stars which however have not been studied systemat…
The symmetries of flow structures are often prescribed by their mechanical instability and geometry. Here, as an example, we present the homotopy of the rotating 3-fold spiral state that is robust in a spherical Couette flow towards the hybrid system with a thermal stratification effect. It has not yet been confirmed that the rotating wave state smoothly connects to the thermal stratification system. Through continuation, the most dangerous mode at a purely spherical Couette flow of $m=4$ modes of spherical harmonics is replaced by $l=4$ and $m=3$ in a purely thermal convective system. For the…
We consider the effect of stratification on systematic, large-scale flows generated in anelastic convection. We present results from three-dimensional numerical simulations of convection in a rotating plane layer in which the angle between the axis of rotation and gravity is allowed to vary. This model is representative of different latitudes of a spherical body. We consider two distinct parameter regimes: (i) weakly rotating and (ii) rapidly rotating. In each case, we examine the effect of stratification on the flow structure and heat transport properties focussing on the difference between B…
We present 3D implicit large eddy simulations (ILES) of the turbulent convection in the envelope of a 5 Msun red giant star and in the oxygen-burning shell of a 23 Msun supernova progenitor. The numerical models are analyzed in the framework of 1D Reynolds-Averaged Navier-Stokes (RANS) equations. The effects of pressure fluctuations are more important in the red giant model, owing to larger stratification of the convective zone. We show how this impacts different terms in the mean-field equations. We clarify the driving sources of kinetic energy, and show that the rate of turbulent dissipation…
We study the dynamical regimes of a density-stratified fluid confined between isothermal no-slip top and bottom boundaries (at temperatures $T_t$ and $T_b$) via direct numerical simulation. The thermal expansion coefficient of the fluid is temperature dependent and chosen such that the fluid density is maximum at the inversion temperature $T_b>T_i>T_t$. Thus, the lower layer of the fluid is convectively unstable while the upper layer is stably stratified. We show that the characteristics of the convection change significantly depending on the degree of stratification of the stable layer. For s…
A water-filled differentially heated rotating annulus with initially prepared stable vertical salinity profiles is studied in the laboratory. Based on two-dimensional horizontal particle image velocimetry (PIV) data, and infrared camera visualizations, we describe the appearance and the characteristics of the baroclinic instability in this original configuration. First, we show that when the salinity profile is linear and confined between two non stratified layers at top and bottom, only two separate shallow fluid layers can be destabilized. These unstable layers appear nearby the top and the …
Double diffusive staircases are investigated experimentally in a fluid layer with a stabilizing temperature gradient and a destabilizing gradient of ion concentration. Gradients of temperature and ion concentration are maintained in a steady state within an electrochemical system. Staircases are observed even if the density stratification is unstable. None of the previously proposed mechanisms for staircase formation can be recognized in the experiments. Ion transport through fingers which are part of a staircase is not the same as transport through fingers extending through the entire cell. F…
The parameterization of small-scale turbulent fluctuations in convective systems and in the presence of strong stratification is a key issue for many applied problems in oceanography, atmospheric science and planetology. In the presence of stratification, one needs to cope with bulk turbulent fluctuations and with inversion regions, where temperature, density -or both- develop highly non-linear mean profiles due to the interactions between the turbulent boundary layer and the unmixed -stable- flow above/below it. We present a second order closure able to cope simultaneously with both bulk and …
We extend the notion of a shell model to stratified systems, and propose one that represents stratified, nonmagnetic, nonrotating convection at low Mach number. Motivated by profiles of background stratification that support convection in stars such as the Sun, we study numerical solutions corresponding to a highly unstable layer above a mildly unstable layer. We find that at low Prandtl number, convective amplitudes decrease with depth in the lower layer. This suggests that the suppression of convection in the deeper layers of the Sun's convection zone (the convective conundrum) can be addres…
Numerical simulations of convection in a layer filled with ideal gas are presented. The control parameters are chosen such that there is a significant variation of density of the gas in going from the bottom to the top of the layer. The relations between the Rayleigh, Peclet and Nusselt numbers depend on the density stratification. It is proposed to use a data reduction which accounts for the variable density by introducing into the scaling laws an effective density. The relevant density is the geometric mean of the maximum and minimum densities in the layer. A good fit to the data is then obt…
We present three-dimensional direct numerical simulations of internal waves excited by turbulent convection in a self-consistent, Boussinesq and Cartesian model of convective--stably-stratified fluids. We demonstrate that in the limit of large Rayleigh number ($Ra\in [4\times 10^7,10^9]$) and large stratification (Brunt-Väisälä frequencies $f_N \gg f_c$, where $f_c$ is the convective frequency), simulations are in good agreement with a theory that assumes waves are generated by Reynolds stresses due to eddies in the turbulent region (Lecoanet \& Quataert 2013 MNRAS 430 (3) 2363-2376). Specific…
The linear theory for rotating compressible convection in a plane layer geometry is presented for the astrophysically-relevant case of low Prandtl number gases. When the rotation rate of the system is large, the flow remains geostrophically balanced for all stratification levels investigated and the classical (i.e., incompressible) asymptotic scaling laws for the critical parameters are recovered. For sufficiently small Prandtl numbers, increasing stratification tends to further destabilise the fluid layer, decrease the critical wavenumber and increase the oscillation frequency of the convecti…
This study offers an explanation of a recently observed effect of destabilization of free convective flows by weak rotation. After studying several models where flows are driven by a simultaneous action of convection and rotation, it is concluded that the destabilization is observed in the cases where centrifugal force acts against main convective circulation. At relatively low Prandtl numbers this counter action can split the main vortex into two counter rotating vortices, where the interaction leads to instability. At larger Prandtl numbers, the counter action of the centrifugal force steepe…
Stably-stratified layers may be present at the top of the electrically-conducting fluid layers of many planets either because the temperature gradient is locally subadiabatic or because a stable composition gradient is maintained by the segregation of chemical elements. Here we study the double-diffusive processes taking place in such a stable layer, considering the case of Mercury's core where the temperature gradient is stable but the composition gradient is unstable over a 800km-thick layer. The large difference in the molecular diffusivities leads to the development of buoyancy-driven inst…
In planetary fluid cores, the density depends on temperature and chemical composition, which diffuse at very different rates. This leads to various instabilities, bearing the name of double-diffusive convection. We investigate rotating double-diffusive convection (RDDC) in fluid spheres. We use the Boussinesq approximation with homogeneous internal thermal and compositional source terms. We focus on the finger regime, in which the thermal gradient is stabilising whereas the compositional one is destabilising. First, we perform a global linear stability analysis in spheres. The critical Rayleig…
Stellar convection is a non-local process responsible for the transport of heat and chemical species. It can lead to enhanced mixing through convective overshooting and excitation of internal gravity waves (IGWs) at convective boundaries. The relationship between these processes is still not well understood and requires global hydrodynamic simulations to capture the important large-scale dynamics. The steep stratification in stellar interiors suggests that the radial extent of such simulations can affect the convection dynamics, the IGWs in the stably stratified radiative zone, and the depth o…
We show that the phase space of stratified turbulence mainly consists of two slow invariant manifolds with rich physics, embedded on a larger basin with fast evolution. A local invariant manifold in the vicinity of the fluid at equilibrium corresponds to waves, while a global invariant manifold corresponds to the onset of local convection. Using a reduced model derived from the Boussinesq equations, we propose that waves accumulate energy nonlinearly up to a point such that fluid elements escape from the local manifold and evolve fast to the global manifold, where kinetic energy can be more ef…
Observations of low-mass stars reveal a variety of magnetic field topologies ranging from large-scale, axial dipoles to more complex magnetic fields. At the same time, three-dimensional spherical simulations of convectively driven dynamos reproduce a similar diversity, which is commonly obtained either with Boussinesq models or with more realistic models based on the anelastic approximation, which take into account the variation of the density with depth throughout the convection zone. Nevertheless, a conclusion from different anelastic studies is that dipolar solutions seem more difficult to …
We develop a theory of three-dimensional slow Rossby waves in rotating spherical density stratified convection. The Rossby waves with frequencies which are much smaller than the rotating frequency, are excited by a non-axisymmetric instability from the equilibrium based on the developed convection. These waves interact with the inertial waves and the density stratified convection. The density stratification is taken into account using the anelastic approximation for very low-Mach-number flows. We study long-term planetary Rossby waves with periods which are larger than two years. We suggest th…
Foundations of a new projection-based model reduction approach for convection dominated nonlinear fluid flows are summarized. In this method the evolution of the flow is approximated in the Lagrangian frame of reference. Global basis functions are used to approximate both the state and the position of the Lagrangian computational domain. It is demonstrated that in this framework, certain wave-like solutions exhibit low-rank structure and thus, can be efficiently compressed using relatively few global basis. The proposed approach is successfully demonstrated for the reduction of several simple …
Chemical reactions can accelerate, slow down or even be at the very origin of the development of dissolution-driven convection in partially miscible stratifications, when they impact the density profile in the host fluid phase. We numerically analyze the dynamics of this reactive convective dissolution in the fully developed non-linear regime for a phase A dissolving into a host layer containing a dissolved reactant B. We show that for a general A+B$\rightarrow$C reaction in solution, the dynamics vary with the Rayleigh numbers of the chemical species, i.e. with the nature of the chemicals in …
We examine the translation of a sphere in a stably stratified ambient in the limit of small Reynolds ($Re \ll 1$) and viscous Richardson numbers ($Ri_v \ll 1$); here, $Re = \frac{ρUa}μ$ and $Ri_v = \frac{γa^3 g}{μU}$ with $a$ being the sphere radius, $U$ the translation speed, $ρ$ and $μ$ the density and viscosity of the stratified ambient, $g$ the acceleration due to gravity, and $γ$ the density gradient (assumed constant) characterizing the ambient stratification. In contrast to most earlier efforts, our study considers the convection dominant limit corresponding to $Pe = \frac{Ua}{D} \gg 1$…
We report a novel form of convection in suspensions of the bioluminiscent marine bacterium $Photobacterium~phosphoreum$. Suspensions of these bacteria placed in a chamber open to the air create persistent luminiscent plumes most easily visible when observed in the dark. These flows are strikingly similar to the classical bioconvection pattern of aerotactic swimming bacteria, which create an unstable stratification by swimming upwards to an air-water interface, but they are a puzzle since the strain of $P.~phosphoreum$ used does not express flagella and therefore cannot swim. Systematic experim…
We study fully compressible convection in the context of plane-parallel, polytropically stratified atmospheres. We perform a suite of 2D and 3D simulations in which we vary the initial superadiabaticity ($ε$) and the Rayleigh number (Ra) while fixing the initial density stratification, aspect ratio, and Prandtl number. The evolved heat transport, quantified by the Nusselt number (Nu), follows scaling relationships similar to those found in the well-studied, incompressible Rayleigh-Bénard problem. This scaling holds up in both 2D and 3D and is not appreciably affected by the magnitude of $ε$.
