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Register a new userTheory of stellar convection: Removing the Mixing-Length Parameter
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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.
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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.
(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.
We present evolutionary models for solar-like stars with an improved treatment of convection that results in a more accurate estimate of the radius and effective temperature. This is achieved by improving the calibration of the mixing-length parameter, which sets the length scale in the 1D convection model implemented in the stellar evolution code. Our calibration relies on the results of 2D and 3D radiation hydrodynamics simulations of convection to specify the value of the adiabatic specific entropy at the bottom of the convective envelope in stars as a function of their effective temperature, surface gravity and metallicity. For the first time, this calibration is fully integrated within the flow of a stellar evolution code, with the mixing-length parameter being continuously updated at run-time. This approach replaces the more common, but questionable, procedure of calibrating the length scale parameter on the Sun, and then applying the solar-calibrated value in modeling other stars, regardless of their mass, composition and evolutionary status. The internal consistency of our current implementation makes it suitable for application to evolved stars, in particular to red giants. We show that the entropy calibrated models yield a revised position of the red giant branch that is in better agreement with observational constraints than that of standard models.
We present here the first stellar models on the Hertzsprung-Russell diagram (HRD), in which convection is treated according to the novel scale-free convection theory (SFC theory) by Pasetto et al. (2014). The aim is to compare the results of the new theory with those from the classical, calibrated mixing-length (ML) theory to examine differences and similarities. We integrate the equations describing the structure of the atmosphere from the stellar surface down to a few percent of the stellar mass using both ML theory and SFC theory. The key temperature over pressure gradients, the energy fluxes, and the extension of the convective zones are compared in both theories. The analysis is first made for the Sun and then extended to other stars of different mass and evolutionary stage. The results are adequate: the SFC theory yields convective zones, temperature gradients of the ambient and of the convective element, and energy fluxes that are very similar to those derived from the calibrated MT theory for main sequence stars. We conclude that the old scale dependent ML theory can now be replaced with a self-consistent scale-free theory able to predict correct results, one which is simpler and more physically grounded than the ML theory. Fundamentally, the SFC theory offers a deeper insight of the underlying physics than numerical simulations.
The helioseismic observations of the internal rotation profile of the Sun raise questions about the two-dimensional (2D) nature of the transport of angular momentum in stars. Here we derive a convective prescription for axisymmetric (2D) stellar evolution models. We describe the small scale motions by a spectrum of unstable linear modes in a Boussinesq fluid. Our saturation prescription makes use of the angular dependence of the linear dispersion relation to estimate the anisotropy of convective velocities. We are then able to provide closed form expressions for the thermal and angular momentum fluxes with only one free parameter, the mixing length. We illustrate our prescription for slow rotation, to first order in the rotation rate. In this limit, the thermodynamical variables are spherically symetric, while the angular momentum depends both on radius and latitude. We obtain a closed set of equations for stellar evolution, with a self-consistent description for the transport of angular momentum in convective regions. We derive the linear coefficients which link the angular momentum flux to the rotation rate ($Lambda$- effect) and its gradient ($alpha$-effect). We compare our results to former relevant numerical work.
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