Extreme value statistics for two-dimensional convective penetration in a pre-main sequence star

Abstract

Context. In the interior of stars, a convectively unstable zone typically borders a zone that is stable to convection. Convective motions can penetrate the boundary between these zones, creating a layer characterized by intermittent convective mixing, and gradual erosion of the density and temperature stratification. Aims. We examine a penetration layer formed between a central radiative zone and a large convection zone in the deep interior of a young low-mass star. Using the Multidimensional Stellar Implicit Code (MUSIC) to simulate two-dimensional compressible stellar convection in a spherical geometry over long times, we produce statistics that characterize the extent and impact of convective penetration in this layer. Methods. We apply extreme value theory to the maximal extent of convective penetration at any time. We compare statistical results from simulations which treat non-local convection, throughout a large portion of the stellar radius, with simulations designed to treat local convection in a small region surrounding the penetration layer. For each of these situations, we compare simulations of different resolution, which have different velocity magnitudes. We also compare statistical results between simulations that radiate energy at a constant rate to those that allow energy to radiate from the stellar surface according to the local surface temperature. Results. Based on the frequency and depth of penetrating convective structures, we observe two distinct layers that form between the convection zone and the stable radiative zone. We show that the probability density function of the maximal depth of convective penetration at any time corresponds closely in space with the radial position where internal waves are excited. We find that the maximal penetration depth can be modeled by a Weibull distribution with a small shape parameter. Using these results, and building on established scalings for diffusion enhanced by large-scale convective motions, we propose a new form for the diffusion coefficient that may be used for one-dimensional stellar evolution calculations in the large Péclet number regime. These results should contribute to the 321D link.

A Model of Rotating Convection in Stellar and Planetary Interiors. I. Convective Penetration

“…This length scale may be interpreted as depth to which the strongest downflows travel before stopping. This behavior is at least roughly consistent with the model proposed here, and with the results of Pratt et al (2017b) and Pratt et al (2017a), although it is in contrast to the models proposed in Freytag et al (1996) and Herwig (2000). For the time being, the examinations in Julien et al (1996) and Brummell et al (2002) will have to suffice as evidence that the over-shooting is reduced with rotation rate, while the shape of the mixing region likely depend upon the model setup and equations solved in similar simulations.…”

“…A frequent assumption made of turbulent flows is to assume that the underlying distribution of a flow quantity is normal. However, it is shown through the simulations and analysis of Pratt et al (2017b) and Pratt et al (2017a) that the more rare events occurring in the tails of the actual distribution function, e.g. the extremal penetrative flows with a higher velocity and a greater entropy deficit than the average flow, have a larger impact than is expected in Gaussian turbulence models on the turbulent mixing coefficients.…”

Probing the shape of the mixing profile and of the thermal structure at the convective core boundary through asteroseismology

“…Another way to parameterise diffusive mixing processes is to use a profile of the mixing coefficient based upon the Gumbel distribution. Pratt et al (2017) simulations of the young Sun. This diffusion coefficient is derived to describe flows in a large Péclet number regime (Pe 1), which is characteristic of stellar interiors.…”

Impact of convective boundary mixing on the TP-AGB

“…This is a fact which can become especially relevant when considering the evolution during the thermally pulsing Asymptotic Giant Branch (TP-AGB), where boundaries at the edge of convective zones are known to play an important E-mail: wagstaff@mpa-garching.mpg.de † E-mail: marcelo@mpa-garching.mpg.de part in governing many important properties and observables of these stars (Herwig et al 1997;Herwig 2005), such as third dredge-up (TDU) and the Initial-Final Mass Relation (IFMR). Certainly, it has been seen in hydrodynamic simulations (Freytag et al 1996;Herwig et al 2007;Meakin & Arnett 2007;Mocák et al 2009;Baraffe et al 2017;Pratt et al 2017) that the strict Schwarzschild boundary, as implemented in 1D stellar evolution codes, simply does not appear in the form of a composition discontinuity in a spherically symmetric manner when it comes to multi-dimensional models.…”