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The solar tachocline is a thin internal boundary layer in the Sun located between the differentially rotating convection zone and the uniformly rotating region of the radiative interior beneath. E. A. Spiegel & J. P. Zahn 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, which is 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 G. P. Chini et al. and K. Shah et al., which has been numerically validated by P. Garaud et al. 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 Ω/N, where Ω is the mean angular velocity of the Sun and N is the characteristic 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. 

Abstract Post-Cassini ring seismology analysis suggests the existence of a stable stratification inside Saturn that extends from the center to ∼60% of its radius, in what is recognized today as Saturn’s dilute core. Similarly, gravity measurements on Jupiter suggest the existence of a dilute core of weekly constrained radial extent. These cores are likely in a double-diffusive regime, which prompts the question of their long-term stability. Indeed, previous direct numerical simulation (DNS) 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 paper, 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 DNSs 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 discoveredγ-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. 

We investigate the linear stability of a sinusoidal shear flow with an initially uniform streamwise magnetic field in the framework of incompressible magnetohydrodynamics (MHD) with finite resistivity and viscosity. This flow is known to be unstable to the Kelvin–Helmholtz instability in the hydrodynamic case. The same is true in ideal MHD, where dissipation is neglected, provided the magnetic field strength does not exceed a critical threshold beyond which magnetic tension stabilizes the flow. Here, we demonstrate that including viscosity and resistivity introduces two new modes of instability. One of these modes, which we refer to as an Alfvénic Dubrulle–Frisch instability, exists for any non-zero magnetic field strength as long as the magnetic Prandtl number $${{{Pm}}} < 1$$ . We present a reduced model for this instability that reveals its excitation mechanism to be the negative eddy viscosity of periodic shear flows described by Dubrulle & Frisch ( Phys. Rev. A, vol. 43, 1991, pp. 5355–5364). Finally, we demonstrate numerically that this mode saturates in a quasi-stationary state dominated by counter-propagating solitons. 

Abstract We study the properties of oscillatory double-diffusive convection (ODDC) in the presence of a uniform vertical background magnetic field. ODDC takes place in stellar regions that are unstable according to the Schwarzschild criterion and stable according to the Ledoux criterion (sometimes called semiconvective regions), which are often predicted to reside just outside the core of intermediate-mass main-sequence stars. Previous hydrodynamic studies of ODDC have shown that the basic instability saturates into a state of weak wave-like convection, but that a secondary instability can sometimes transform it into a state of layered convection, where layers then rapidly merge and grow until the entire region is fully convective. We find that magnetized ODDC has very similar properties overall, with some important quantitative differences. A linear stability analysis reveals that the fastest-growing modes are unaffected by the field, but that other modes are. Numerically, the magnetic field is seen to influence the saturation of the basic instability, overall reducing the turbulent fluxes of temperature and composition. This in turn affects layer formation, usually delaying it, and occasionally suppressing it entirely for sufficiently strong fields. Further work will be needed, however, to determine the field strength above which layer formation is actually suppressed in stars. Potential observational implications are briefly discussed. 

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 are 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.