Non-local Convection Theory Research Articles

Seismic study of solar convection and overshooting: results of nonlocal convection

Local mixing-length theory is incapable of describing nonlocal phenomena in stellar convection, such as overshooting. Therefore standard solar models constructed with local mixing-length theory significantly deviate from the Sun at the boundariesof the convection zone, where convection becomes less efficient and nonlocal effects are important. The differences between observed and computed frequencies mainly come from the region near the surface, while the localized difference in sound speed is just below the convective envelope. We compute a solar envelope model using Xiong's nonlocal convection theory, and carry out helioseismic analysis. The nonlocal model has a smooth transition at the base of the convection zone, as revealed by helioseismology. It reproduces solar frequencies more accurately, and reduces the localized difference in sound speed between the Sun and standard solar models.

SOUND-SPEED INVERSION OF THE SUN USING A NONLOCAL STATISTICAL CONVECTION THEORY

Helioseismic inversions reveal a major discrepancy in sound speed between the Sun and the standard solar model just below the base of solar convection zone. We demonstrate that this discrepancy is caused by the inherent shortcomings of the local mixing-length theory adopted in the standard solar model. Using a self-consistent nonlocal convection theory, we construct an envelope model of the Sun for sound-speed inversion. Our solar model has a very smooth transition from the convective envelope to the radiative interior; and the convective energy flux changes sign crossing the boundaries of the convection zone. It shows evident improvement over the standard solar model, with a significant reduction in the discrepancy in sound speed between the Sun and local convection models.

A Note on the Supersonic Convection in Stellar Atmospheres

By using a non-local convection theory, both the local and nonlocal convective envelope models of evolutionary series of stars with masses from 1 to 30 solar masses are calculated. The problem of supersonic convection is reviewed. The results show that the convective velocities in the stellar atmosphere are seriously overestimated by the local mixing-length theory. Convection is strongly supersonic in the atmospheres of yellow giant and super-giants, while the local mixing-length theory is used. However, it becomes subsonic for most stars when convection returns to the normal nonlocal treatment. Convection velocities increase with increase of luminosities of stars. There is still weak supersonic convection in few red and yellow giant and super-giants. It is suspected whether this supersonic convection in stellar atmospheres is true.

Lithium depletion in late-type dwarfs

Using a self-consistent dynamic theory of non-local convection, the atmospheric lithium-abundance depletion values of a series of stellar evolutionary models with M = 0.725-1.5 M(circle dot) are calculated, and the results show that the general observed properties of lithium abundance in lower-main-sequence stars can be reproduced by overshooting mixing and gravitational settling. After a careful study of the mechanisms of lithium depletion in stars, it is concluded that overshooting mixing and microdiffusion induced by gravitational settling and radiative acceleration are the two primary depletion mechanisms: for warm stars with M = 1.1 M(circle dot) (or T(e) = 6100 K) microdiffusion dominates, while it is the other way round for cooler objects with M <= 1 M(circle dot) (or T(e) <= 5800 K).

How to define the boundaries of a convective zone, and how extended is overshooting?

In non-local convection theory, convection extends without limit and therefore an apparent boundary cannot be defined clearly, as in local theory. From the requirement that a similar structure for both local and non-local models has the same depth of convection zone, and taking into account the driving mechanism of turbulent convection, we argue that a proper definition of the boundary of a convective zone should be the place where the convective energy flux (i.e. the correlation of turbulent velocity and temperature) changes its sign. Therefore, it is a convectively unstable region when the flux is positive, and it is a convective overshooting zone when the flux becomes negative. The physical picture of the overshooting zone drawn by the usual non-local mixing-length theory is incorrect. In fact, convection is already subadiabatic (del < del(ad)) long before reaching the unstable boundary, while in the overshooting zone below the convective zone, convection is subadiabatic and superradiative (del(rad) < del < del(ad)). The transition between the adiabatic and radiative temperature gradients is continuous and smooth instead of being a sudden switch. In the unstable zone, the temperature gradient approaches a radiative temperature gradient rather than an adiabatic temperature gradient. We would like to note again that the overshooting distance is different for different physical quantities. In an overshooting zone at deep stellar interiors, the e-folding lengths of turbulent velocity and temperature are about 0.3H(P), whereas that of the velocity-temperature correlation is much shorter, about 0.09H(P). The overshooting distance in the context of stellar evolution, measured by the extent of the mixing of stellar matter, should be more extended. It is estimated to be as large as 0.25-1.7H(P) depending on the evolutionary time-scale. The larger the overshooting distance, the longer the time-scales. This is because of the participation of the extended overshooting tail in the mixing process.

On the boundaries of a convective zone and the extent of overshooting

AbstractIn this work, we will show that a proper definition of the boundary of a convective zone should be the place where the convective energy flux (i.e. the correlation of turbulent velocity and temperature) changes its sign. Therefore, it is convectively unstable region when the flux is positive, and it is convective overshooting zone when the flux becomes negative. In our nonlocal convection theory, convection is already sub-adiabatic (∇ &lt; ∇ad) far before reaching the unstable boundary; while in the overshooting zone below the convective zone, convection is sub-adiabatic and super-radiative (∇rad &lt; ∇ &lt; ∇ad). The transition between the adiabatic temperature gradient and the radiative one is continuous and smooth instead of a sudden switch. In the unstable zone, the temperature gradient is approaching radiative rather than going to adiabatic. The distance of convective overshooting is different for different physical quantities. The overshooting distance in the context of stellar evolution, measured by the extent of mixing of stellar matter, should be more extended than that of other physical quantities. It is estimated as large as 0.25–1.7 Hp depending on the evolutionary timescale.

Lithium Depletions in Late-type Dwarfs

AbstractUsing a self-consistant dynamic theory of non-local convection in chemically inhomogeneous stars, the lithium depletion in MS stars of masses 0.725–1.5 M⊙ are calculated. Both of the overshooting and microdiffusion are included in a consistent way. The comparisons of theretical reasults with the observed Li depletions in open clusters show that the general characters of Li depletions can be reproduced by theory. The overshooting mixing and microdiffusion induced by gravitational setting and radiative accelerations may be two main mechanisms of Li depletion.

Inclusion of a time-dependent, non-local convection theory in a stellar evolution code

AbstractThe non-local, time-dependent convection theory of Kuhfuß (1986) in both its one- and three-equation form has been implemented in the Garching stellar evolution code. We present details of the implementation and the difficulties encountered. Specific test cases have been calculated, among them a 5 M⊙ star and the Sun. These cases point out deficits of the theory. In particular, the assumption of an isotropic velocity field leads to too extensive overshooting and has to be modified at convective boundaries. Some encouraging aspects are indicated as well.

Core Overshoot: An Improved Treatment and Constraints from Seismic Data

We present a comprehensive set of stellar evolution models for Procyon A in an effort to guide future measurements of both traditional stellar parameters and seismic frequencies toward constraining the amount of core overshoot in Procyon A and possibly other stars. Current observational measurements of Procyon A when combined with traditional stellar modeling only place a large upper limit on overshoot of αOV < 1.1. By carrying out a detailed pulsation analysis, we further demonstrate how p- and g-mode averaged spacings can be used to gain better estimates of the core size. For both p- and g-modes, the frequency spacings for models without overshoot are clearly separated from the models with overshoot. In addition, measurements of the l = 0 averaged small p-mode spacings could be used to establish Procyon A's evolutionary stage. For a fixed implementation of overshoot and under favorable circumstances, the g-mode spacings can be used to determine the overshoot extent to an accuracy of ±0.05HP. However, we stress that considerable confusion is added due to the unknown treatment of the overshoot region. This ambiguity might be removed by analyzing many different stars. A simple nonlocal convection theory developed by Kuhfuss is implemented in our stellar evolution code and contrasted with the traditional approaches. We show that this theory supports a moderate increase of the amount of convective overshoot with stellar mass of ΔαOV +0.10 between 1.5 and 15 M☉. This theory places an upper limit on Procyon A's core overshoot extent of ~0.4HP, which matches the limit imposed by Roxburgh's integral criterion.

Beryllium depletion in the solar atmosphere

Using a non-local convection theory for chemically inhomogeneous stars, the beryllium depletion of the Sun is calculated. The results show that Be is slightly depleted (∼5%) in the evolutionary model for the Sun.

The extent of the solar overshooting zone deduced from Li abundance

Using a non-local convection theory for chemically inhomogeneous stars, a set of models of the solar convective envelope is presented in this work. The (7) Li depletion as a result of convective overshooting is calculated. A model for the solar convective envelope is found which meets the observational constraints from both the Li abundance and depth of the convection zone determined by helioseismology. We argue that there is no general overshooting distance that holds for all physical quantities, and the differences between them can be factors of a few. In the case of solar lithium depletion, the effective overshooting distance at the low convective boundary can be extended to 1.07H(P) (where H-P is the local pressure scaleheight at the boundary), while the corresponding super-radiative temperature zone overshoots to 0.2H(P) in the sense of helioseismology, and the e-folding length for turbulent velocity and temperature is about 0.38H(P) for this solar model.

Lithium depletion in the solar atmosphere

Using a complete non-local convection theory, we carried out the theoretical calculations of 7 Li depletion of the solar convective envelope models with different convective parameters c 1 and c 2, and got a model of the solar convection zone consistent with the observed 7 Li abundance and the depth of the solar convection zone determined by helioseismic techniques. The overshooting distance of effective non-local convective mixing of 7 Li is very extensive, which is about 1.07 H P or 0.09 R ⊙. However, the super-radiative temperature zone is much narrower, and it is only 0.20 H P or 0.016 R ⊙ .

The middle-high degree adiabatic p-mode eigenfrequencies of the solar envelope models

In this paper, the adiabatic p-mode eigenfrequencies for five solar convective envelope models are calculated, using Xiong's nonlocal convection theory. At the precision level of one percent, they are well consistent with observation. From the plotting of Δν(= ν obs − ν cal) vs. ν, it is obvious for the models with smaller convection parameter c 1 (the models I and II), that the frequency differences fall into two distinct groups. For the modes with l > 60, the differencies decrease as the frequency increases, distributing in an inclined narrow strip. This indicates that the main error source is located in the out- ermost 5% R ⊙ region of the models, possibly due to the nonadiabatic effects and the uncertainty in the equation of state. For the modes with l ⩽ 60, the theoretical eigenfrequencies are systematically smaller than the observed ones, implying that another error source locates at the bottom of the solar convection zone and the overshooting region below. It may be most likely interpreted by the existence of a magnetic field of 10 3– 10 4 gauss in the lower part of the solar convection zone.

Stellar Convection

AbstractTime-dependent convection theory and non-local convection theory are described. The difference between the convection theories results from the different treatment of the non-linear terms of the hydrodynamic equations. We have obtained a better understanding of the thermodynamic coupling between convection and oscillations in comparison to the dynamic coupling.

A statistical theory of non-local convection

Existing problems in the theory of local convection are analysed. It is shown that the effect of turbulent pressure on the structure and vibrational stability of stars with extensive convective envelopes is not negliable. The physical significance of Henyey's numerical instability in calculations of convective envelopes is elucidated. When the effect of turbulent pressure on hydrostatic equilibrium of stars is considered, we have to abandon the local treatment of convection. We propose a theory of non-local convection based on the dynamic equations of correlation functions.