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Register a new userConvection with Misaligned Gravity and Rotation: Simulations and Rotating Mixing Length Theory
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We present numerical simulations, using two complementary setups, of rotating Boussinesq thermal convection in a three-dimensional Cartesian geometry with misaligned gravity and rotation vectors. This model represents a small region at a non-polar latitude in the convection zone of a star or planet. We investigate the effects of rotation on the bulk properties of convection at different latitudes, focusing on determining the relation between the heat flux and temperature gradient. We show that our results may be interpreted using rotating mixing length theory (RMLT). The simplest version of RMLT (due to Stevenson) considers the single mode that transports the most heat. This works reasonably well in explaining our results, but there is a systematic departure from these predictions (up to approximately $30%$ in the temperature gradient) at mid-latitudes. We develop a more detailed treatment of RMLT that includes the transport afforded by multiple modes, and we show that this accounts for most of the systematic differences. We also show that convectively-generated zonal flows and meridional circulations are produced in our simulations, and that their properties depend strongly on the dimensions of the box. These flows also affect the heat transport, contributing to departures from RMLT at some latitudes. However, we find the theoretical predictions of the multi-mode theory for the mid-layer temperature gradient, the root-mean-square (RMS) vertical velocity, the RMS temperature fluctuation, and the spatial spectrum of the heat transport at different latitudes, are all in reasonably good agreement with our numerical results when zonal flows are small.
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(abridged) The calculation of the thermal stratification in the superadiabatic layers of stellar models with convective envelopes is a long standing problem of stellar astrophysics, and has a major impact on predicted observational properties like radius and effective temperature. The Mixing Length Theory, almost universally used to model the superadiabatic convective layers, contains effectively one free parameter to be calibrated --alpha(ml)-- whose value controls the resulting effective temperature. Here we present the first self-consistent stellar evolution models calculated by employing the atmospheric temperature stratification, Rosseland opacities, and calibrated variable alpha(ml) (dependent on effective temperature and surface gravity) from a large suite of three-dimensional radiation hydrodynamics simulations of stellar convective envelopes and atmospheres for solar stellar composition (Trampedach et al. 2013). From our calculations (with the same composition of the radiation hydrodynamics simulations), we find that the effective temperatures of models with the hydro-calibrated variable alpha(ml) display only minor differences, by at most ~30-50 K, compared to models calculated at constant solar alpha(ml). The depth of the convective regions is essentially the same in both cases. We have also analyzed the role played by the hydro-calibrated T(tau) relationships in determining the evolution of the model effective temperatures, when compared to alternative T(tau) relationships often used in stellar model computations. The choice of the T(tau) can have a larger impact than the use of a variable alpha(ml) compared to a constant solar value. We found that the solar semi-empirical T(tau) by Vernazza et al. (1981) provides stellar model effective temperatures that agree quite well with the results with the hydro-calibrated relationships.
The mixing of a passive scalar like lithium, beryllium or temperature fluctuations due to the magnetic Tayler instability of a rotating axial pinch is considered. Our study is carried out within a Taylor-Couette setup for two rotation laws: quasi-Kepler and solid-body rotation. The minimum magnetic Prandtl number used is 0.05 while the molecular Schmidt number Sc of the fluid varies between 0.1 and 2. An effective diffusivity coefficient for the mixing is numerically measured by the decay process of a global concentration peak located between the cylinder walls. We find that only models with Sc>0.1 do provide finite eddy diffusivity values. We also find that for quasi-Kepler rotation at a magnetic Mach number Mm~2 the flow transits from the slow-rotation regime to the fast-rotation regime. For fixed Reynolds number the relation between the normalized eddy diffusivity and the Schmidt number of the fluid is always linear so that also a linear relation between the instability-induced diffusivity and the molecular viscosity results just in the sense proposed by Schatzman (1977). The numerical value of the coefficient in this relation will reach a maximum at Mm~2 and will decrease for Mm>>1 implying that only toroidal magnetic fields of order kG can exist in the solar tachocline.
Stellar convection is customarily described by Mixing-Length Theory, which makes use of the mixing-length scale to express the convective flux, velocity, and temperature gradients of the convective elements and stellar medium. The mixing-length scale is taken to be proportional to the local pressure scale height, and the proportionality factor (the mixing-length parameter) must be determined by comparing the stellar models to some calibrator, usually the Sun. No strong arguments exist to suggest that the mixing-length parameter is the same in all stars and at all evolutionary phases. The aim of this study is to present a new theory of stellar convection that does not require the mixing length parameter. We present a self-consistent analytical formulation of stellar convection that determines the properties of stellar convection as a function of the physical behaviour of the convective elements themselves and of the surrounding medium. This new theory is formulated starting from a conventional solution of the Navier-Stokes/Euler equations, i.e. the Bernoulli equation for a perfect fluid, but expressed in a non-inertial reference frame co-moving with the convective elements. In our formalism the motion of stellar convective cells inside convectively-unstable layers is fully determined by a new system of equations for convection in a non-local and time-dependent formalism. We obtain an analytical, non-local, time-dependent sub-sonic solution for the convective energy transport that does not depend on any free parameter. The theory is suitable for the outer convective zones of solar type stars and stars of all mass on the main sequence band. The predictions of the new theory are compared with those from the standard mixing-length paradigm for the most accurate calibrator, the Sun, with very satisfactory results.
Turbulent properties of the quiet Sun represent the basic state of surface conditions, and a background for various processes of solar activity. Therefore understanding of properties and dynamics of this `basic state is important for investigation of more complex phenomena, formation and development of observed phenomena in the photosphere and atmosphere. For characterization of the turbulent properties we compare kinetic energy spectra on granular and sub-granular scales obtained from infrared TiO observations with the New Solar Telescope (Big Bear Solar Observatory) and from 3D radiative MHD numerical simulations (SolarBox code). We find that the numerical simulations require a high spatial resolution with 10 - 25 km grid-step in order to reproduce the inertial (Kolmogorov) turbulence range. The observational data require an averaging procedure to remove noise and potential instrumental artifacts. The resulting kinetic energy spectra show a good agreement between the simulations and observations, opening new perspectives for detailed joint analysis of more complex turbulent phenomena on the Sun, and possibly on other stars. In addition, using the simulations and observations we investigate effects of background magnetic field, which is concentrated in self-organized complicated structures in intergranular lanes, and find an increase of the small-scale turbulence energy and its decrease at larger scales due to magnetic field effects.
We continue our investigation into the nonlinear evolution of the Goldreich-Schubert-Fricke (GSF) instability in differentially rotating radiation zones. This instability may be a key player in transporting angular momentum in stars and giant planets, but its nonlinear evolution remains mostly unexplored. In a previous paper we considered the equatorial instability, whereas here we simulate the instability at a general latitude for the first time. We adopt a local Cartesian Boussinesq model in a modified shearing box for most of our simulations, but we also perform some simulations with stress-free, impenetrable, radial boundaries. We first revisit the linear instability and derive some new results, before studying its nonlinear evolution. The instability is found to behave very differently compared with its behaviour at the equator. In particular, here we observe the development of strong zonal jets (layering in the angular momentum), which can considerably enhance angular momentum transport, particularly in axisymmetric simulations. The jets are, in general, tilted with respect to the local gravity by an angle that corresponds initially with that of the linear modes, but which evolves with time and depends on the strength of the flow. The instability transports angular momentum much more efficiently (by several orders of magnitude) than it does at the equator, and we estimate that the GSF instability could contribute to the missing angular momentum transport required in both red giant and subgiant stars. It could also play a role in the long-term evolution of the solar tachocline and the atmospheric dynamics of hot Jupiters.
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