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 viscosity and the turbulent diffusivity of a passive tracer 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 (JPr) c for instability, adequately accounts for the transition to turbulence in the flow, with (JPr) c 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. 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.

Giant planets like Jupiter and Saturn feature strong zonal wind patterns on their surfaces. Although several different mechanisms that may drive these jets have been proposed over the last decades, the origin of the zonal winds is still unclear. Here, we explore the possibility that the interplay of planetary rotation with the compression and expansion of the convecting fluid can drive multiple deep zonal jets by a compressional Rhines-type mechanism, as originally proposed by Ingersoll and Pollard (1982). In a certain limit, this deep mechanism is shown to be mathematically analogous to the classical Rhines mechanism possibly operating at cloud level. Jets are predicted to occur on a compressional Rhines lengthwhere Ω is the angular velocity, H −1 ρ is the mean inverse density scale height and v jet is the typical jet velocity. Two-dimensional numerical simulations using the anelastic approximation reveal that this mechanism robustly generates jets of the predicted width, and that it typically dominates the dynamics in systems deeper than O(l R ). Potential vorticity staircases are observed to form spontaneously and are typically accompanied by unstably stratified buoyancy staircases. The mechanism only operates at large rotation rates, exceeding those typically reached in three-dimensional simulations of deep convection in spherical shells. Applied to Jupiter and Saturn, the compressional Rhines scaling reasonably fits the available observations. Interestingly, even weak vertical density variations such as those in the Earth core can give rise to a large number of jets, leading to fundamentally different flow structures than predicted by the Boussinesq models typically used in this context.

Numerical simulations of turbulent Rayleigh-Bénard convection in an ideal gas, using either the anelastic approximation or the fully compressible equations, are compared. Theoretically, the anelastic approximation is expected to hold in weakly superadiabatic systems with = ∆T /T r 1, where ∆T denotes the superadiabatic temperature drop over the convective layer and T r the bottom temperature. Using direct numerical simulations, a systematic comparison of anelastic and fully compressible convection is carried out. With decreasing superadiabaticity , the fully compressible results are found to converge linearly to the anelastic solution with larger density contrasts generally improving the match. We conclude that in many solar and planetary applications, where the superadiabaticity is expected to be vanishingly small, results obtained with the anelastic approximation are in fact more accurate than fully compressible computations, which typically fail to reach small for numerical reasons. On the other hand, if the astrophysical system studied contains ∼ O(1) regions, such as the solar photosphere, fully compressible simulations have the advantage of capturing the full physics. Interestingly, even in weakly superadiabatic regions, like the bulk of the solar convection zone, the errors introduced by using artificially large values for for efficiency reasons remain moderate. If quantitative errors of the order of 10% are acceptable in such low regions, our work suggests that fully compressible simulations can indeed be computationally more efficient than their anelastic counterparts.

[1] Magma chambers can be considered as thermochemically driven convection systems. We present a new numerical method that describes the movement of crystallized minerals in terms of active spherical particles in a convecting magma that is represented by an infinite Prandtl number fluid. The main part focuses on the results we obtained. A finite volume thermochemical convection model for two and three dimensions and a discrete element method, which is used to model granular material, are combined. The new model is validated with floating experiments using particles of different densities and an investigation of single and multiparticle settling velocities. The resulting velocities are compared with theoretical predictions by Stokes's law and a hindered settling function for the multiparticle system. Two fundamental convection regimes are identified in the parameter space that is spanned by the Rayleigh number and the chemical Rayleigh number, which is a measure for the density of the particles. We define the T regime that is dominated by thermal convection. Here the thermal driving force is strong enough to keep all particles in suspension. As the particles get denser, they start settling to the ground, which results in a C regime. The C regime is characterized by the existence of a sediment layer with particle-rich material and a suspension layer with few particles. It is shown that the presence of particles can reduce the vigor of thermal convection. In the frame of a parameter study we discuss the change between the regimes that is systematically investigated. We show that the so-called TC transition fits a power law. Furthermore, we investigate the settling behavior of the particles in vigorous thermal convection, which can be linked to crystal settling in magma chambers. We develop an analytical settling law that describes the number of settled particles against time and show that the results fit the observations from numerical and laboratory experiments.Components: 8562 words, 18 figures.

The validity of the anelastic approximation has recently been questioned in the regime of rapidly-rotating compressible convection in low Prandtl number fluids (Calkins, Julien and Marti, Proc. R. Soc. A, 2015, vol. 471, 20140689). Given the broad usage and the high computational efficiency of sound-proof approaches in this astrophysically relevant regime, this paper clarifies the conditions for a safe application. The potential of the alternative pseudoincompressible approximation is investigated, which in contrast to the anelastic approximation is shown to never break down for predicting the point of marginal stability. Its accuracy, however, decreases close to the parameters corresponding to the failure of the anelastic approach, which is shown to occur when the sound-crossing time of the domain exceeds a rotation time scale, i.e. for rotational Mach numbers greater than one. Concerning the supercritical case, which is naturally characterised by smaller rotational Mach numbers, we find that the anelastic approximation does not show unphysical behaviour. Growth rates computed with the linearised anelastic equations converge toward the corresponding fully compressible values as the Rayleigh number increases. Likewise, our fully nonlinear turbulent simulations, produced with our fully compressible and anelastic models and carried out in a highly supercritical, rotating, compressible, low Prandtl number regime show good agreement. However, this nonlinear test example is for only a moderately low convective Rossby number of 0.14. ARTICLE HISTORY