# Turbulent convection model in the overshooting region: II. Theoretical
analysis

Q.S. Zhang11affiliation: National Astronomical Observatories/Yunnan
Observatory, Chinese Academy of Sciences, P.O. Box 110, Kunming 650011, China.
22affiliation: Laboratory for the Structure and Evolution of Celestial
Objects, CAS. 33affiliation: Graduate School of Chinese Academy of Sciences,
Beijing 100039, China. and Y. Li11affiliation: National Astronomical
Observatories/Yunnan Observatory, Chinese Academy of Sciences, P.O. Box 110,
Kunming 650011, China. 22affiliation: Laboratory for the Structure and
Evolution of Celestial Objects, CAS. zqs@ynao.ac.cn(QSZ); ly@ynao.ac.cn(YL)

###### Abstract

Turbulent convection models are thought to be good tools to deal with the
convective overshooting in the stellar interior. However, they are too complex
to be applied in calculations of stellar structure and evolution. In order to
understand the physical processes of the convective overshooting and to
simplify the application of turbulent convection models, a semi-analytic
solution is necessary. We obtain the approximate solution and asymptotic
solution of the turbulent convection model in the overshooting region, and
find some important properties of the convective overshooting: I. The
overshooting region can be partitioned into three parts: a thin region just
outside the convective boundary with high efficiency of turbulent heat
transfer, a power law dissipation region of turbulent kinetic energy in the
middle, and a thermal dissipation area with rapidly decreasing turbulent
kinetic energy. The decaying indices of the turbulent correlations $k$,
$\overline{u_{r}^{\prime}T^{\prime}}$, and $\overline{T^{\prime}T^{\prime}}$
are only determined by the parameters of the TCM, and there is an equilibrium
value of the anisotropic degree $\omega$. II. The overshooting length of the
turbulent heat flux $\overline{u_{r}^{\prime}T^{\prime}}$ is about
$1H_{k}$($H_{k}=|\frac{dr}{dlnk}|$). III. The value of the turbulent kinetic
energy at the convective boundary $k_{C}$ can be estimated by a method called
the maximum of diffusion. Turbulent correlations in the overshooting region
can be estimated by using $k_{C}$ and exponentially decreasing functions with
the decaying indices.

convection — diffusion — turbulence

##  1 Introduction

Convective overshooting is an important physical process in the stellar
structure and evolution. Phenomenologically, the acceleration of a fluid
element is zero at the convective boundary, but its speed is not zero. It is
able to go across the convective boundary into the dynamically stable zone.
This phenomenon is called the convective overshooting. The convective
overshooting transports heat and matter, and affects the structure and
evolution of stars. A phenomenological theory of the overshooting was
developed by Zahn(1991), which predicts an adiabatic overshooting region.
However, Xiong & Deng(2001) pointed out that the turbulent velocity and the
temperature are strongly correlated in Zahn’s theory. Recently, Christensen-
Dalsgaard et al.(2011) found that the convective overshooting only described
by the turbulent convection models could be in agreement with the helioseismic
data.

The turbulent convection models (TCMs) are based on fully hydrodynamic moment
equations, and applied on investigating the convective overshooting(Xiong,
1981, 1985, 1989; Xiong & Deng, 2001; Canuto, 1997; Canuto & Dubovikov, 1998;
Canuto, 1998, 1999; Marik & Petrovay, 2002; Deng & Xiong, 2006; Li & Yang,
2007; Deng & Xiong, 2008; Zhang & Li, 2009). There are two main difficulties
restricting the applications of the TCMs. One is to solve the equations of the
TCMs, which are highly non-linear and unstable in numerical calculations. The
other is to incorporate the TCMs into a stellar evolution code. In general,
solving the TCMs needs the parameters of the stellar structure(e.g.
temperature $T$, density $\rho$, pressure $P$, radius $r$, luminosity $L$, and
elements abundance vector), and solving the equations of stellar structure
requires the temperature gradient $\nabla$ which is determined by the TCMs.
Thus, in order to apply the TCMs, one must solve both the TCMs and the
equations of stellar structure, which shows enormous difficulty. Although
developing numerical technique is very important, getting an approximate
solution of the TCMs is more interesting because an approximate solution helps
to understand the physical processes and may significantly simplify the
application of the TCMs. Xiong(1989) found the asymptotic solution of his TCM
in the overshooting region, the turbulent correlations being exponentially
decreasing in the overshooting region. However, his solution of the heat flux
$\overline{u_{r}^{\prime}T^{\prime}}$ is not suitable near the convective
boundary, and the initial turbulent kinetic energy $k_{0}$ is unknown so that
the value of the turbulent correlations in the overshooting region actually
can not be determined without numerical calculations.

In this paper, we investigate the properties of the convective overshooting by
analyzing Li & Yang’s TCM(Li & Yang, 2007), which was tested in the solar
convection zone(Li & Yang, 2007; Yang & Li, 2007). We try to get a semi-
analytical solution of the TCM in the overshooting region. We introduce the
TCM in Section 2, investigate the properties of the overshooting in Section 3,
and summarize the conclusions in Section 4.

##  2 Turbulent Convection Model

The closure assumptions of Li & Yang’s TCM are(Li & Yang, 2001, 2007): the
three-order moment terms are modeled with a gradient-type scheme; the
dissipation rate $\varepsilon$ of the turbulent kinetic energy $k$ is assumed
to be local; the dissipation rates of the turbulent heat flux
$\overline{u_{r}^{\prime}T^{\prime}}$ and the turbulent fluctuation of
temperature $\overline{T^{\prime}T^{\prime}}$ are assumed to be determined by
both the reciprocal timescale of the turbulent dissipation
$\tau_{1}^{-1}=\frac{\varepsilon}{k}$ and the thermal dissipation one
$\tau_{2}^{-1}=\frac{\lambda}{\rho c_{P}}\frac{\varepsilon^{2}}{k^{3}}$.
According to those closure assumptions, fully hydrodynamic moment equations on
the quasi-steady approximation result in the complete equations of two-order
moment terms(Li & Yang, 2007):

$\displaystyle\frac{1}{\rho r^{2}}\frac{\partial}{\partial r}\left(C_{s}\rho
r^{2}\frac{k}{\varepsilon}\overline{u_{r}^{\prime}u_{r}^{\prime}}\frac{\partial\overline{u_{r}^{\prime}u_{r}^{\prime}}}{\partial
r}\right)=\frac{2}{3}\varepsilon+\frac{2\beta
g_{r}}{T}\overline{u_{r}^{\prime}T^{\prime}}+C_{k}\frac{\varepsilon}{k}\left(\overline{u_{r}^{\prime}u_{r}^{\prime}}-\frac{2}{3}k\right)$
(1) $\displaystyle\frac{1}{\rho r^{2}}\frac{\partial}{\partial
r}\left(C_{s}\rho
r^{2}\frac{k}{\varepsilon}\overline{u_{r}^{\prime}u_{r}^{\prime}}\frac{\partial
k}{\partial r}\right)=\varepsilon+\frac{\beta
g_{r}}{T}\overline{u_{r}^{\prime}T^{\prime}}$ (2) $\displaystyle\frac{2}{\rho
r^{2}}\frac{\partial}{\partial r}\left(C_{t1}\rho
r^{2}\frac{k}{\varepsilon}\overline{u_{r}^{\prime}u_{r}^{\prime}}\frac{\partial\overline{u_{r}^{\prime}T^{\prime}}}{\partial
r}\right)=-\frac{T}{H_{P}}(\nabla-\nabla_{ad})\overline{u_{r}^{\prime}u_{r}^{\prime}}+\frac{\beta
g_{r}}{T}\overline{T^{\prime}T^{\prime}}+C_{t}\left(\frac{\varepsilon}{k}+\frac{\lambda}{\rho
c_{P}}\frac{\varepsilon^{2}}{k^{3}}\right)\overline{u_{r}^{\prime}T^{\prime}}$
(3) $\displaystyle\frac{1}{\rho r^{2}}\frac{\partial}{\partial
r}\left(C_{e1}\rho
r^{2}\frac{k}{\varepsilon}\overline{u_{r}^{\prime}u_{r}^{\prime}}\frac{\partial\overline{T^{\prime}T^{\prime}}}{\partial
r}\right)=-\frac{2T}{H_{P}}(\nabla-\nabla_{ad})\overline{u_{r}^{\prime}T^{\prime}}+2C_{e}\left(\frac{\varepsilon}{k}+\frac{\lambda}{\rho
c_{P}}\frac{\varepsilon^{2}}{k^{3}}\right)\overline{T^{\prime}T^{\prime}}$ (4)

The temperature gradient is calculated as:

$\displaystyle\nabla=\nabla_{R}-\frac{H_{P}}{T}\frac{\rho
c_{P}}{\lambda}\overline{u_{r}^{\prime}T^{\prime}}$ (5)

The meaning of those equations and each term in them were described in
previous works(Li & Yang, 2007; Zhang & Li, 2009) in detail. We simply
introduce them here:

Equations (1-4) describe the equilibrium(time-independent) structure of the
radial kinetic energy $\overline{u_{r}^{\prime}u_{r}^{\prime}}$, the turbulent
kinetic energy $k$, the turbulent heat flux
$\overline{u_{r}^{\prime}T^{\prime}}$ and the turbulent fluctuation of
temperature $\overline{T^{\prime}T^{\prime}}$, respectively. On the left side
of those equations, there is the non-local term(i.e. the diffusion term) of
each turbulent correlation. On the right side, there are the local terms which
describe the generation and the dissipation of each turbulent correlation.

In Eq.(1) and (2), $\varepsilon$ is the turbulent dissipation rate of $k$ and
$\varepsilon=\frac{k^{\frac{3}{2}}}{l}$ where $l=\alpha H_{P}$, and the second
term on the right side is the generation rate of the kinetic energy due to the
contribution of the buoyancy. The last term in Eq.(1) is the return to
isotropy term which attempts to make the turbulent motion be isotropic. In
Eq.(3), the first two terms on the right side is the generation rate of the
turbulent heat flux $\overline{u_{r}^{\prime}T^{\prime}}$, and the last one is
the dissipation rate that comprises the turbulent dissipation and the thermal
dissipation. In Eq.(4), the first term on the right side is the generation
rate of the turbulent fluctuation of temperature
$\overline{T^{\prime}T^{\prime}}$, and the last one is the dissipation rate.
Meanings of other symbols are: $H_{P}=-\frac{dr}{dlnP}$ is the local pressure
scale height, $\beta=-(\frac{\partial ln\rho}{\partial lnT})_{P}$ the
expansion coefficient, $g_{r}=-\frac{GM_{r}}{r^{2}}$ the radial component of
gravity acceleration, $\nabla=\frac{dlnT}{dlnP}$ the temperature gradient in
the stellar interior, $\nabla_{ad}=(\frac{\partial lnT}{\partial lnP})_{S}$
the adiabatic temperature gradient, $\lambda=\frac{4acT^{3}}{3\kappa\rho}$ the
thermal conduction coefficient, $c_{P}=(\frac{\partial H}{\partial T})_{P}$
the specific heat, $C_{k}$ the parameter of the return to isotropy term,
($C_{s},C_{t1},C_{e1}$) the diffusion parameters and ($\alpha,C_{t},C_{e}$)
the dissipation parameters of turbulent
variations($k,\overline{u_{r}^{\prime}T^{\prime}},\overline{T^{\prime}T^{\prime}}$).

In Eqs.(1-4), overbars are only used in three turbulent correlations
$\overline{u_{r}^{\prime}u_{r}^{\prime}}$,
$\overline{u_{r}^{\prime}T^{\prime}}$ and $\overline{T^{\prime}T^{\prime}}$.
The other variations(density $\rho$ and the temperature $T$, etc.) are all
mean state quantities which should use overbars but we ignore them for
convenience.

Equation (5) describes the energy transport in the stellar interior by both
turbulent motions(i.e. convection and overshooting) and radiation.
$\nabla_{R}$ is the radiative temperature gradient.

##  3 Theoretical analysis of TCM in the overshooting region

In the previous work (Zhang & Li, 2009), we applied the TCM in the solar
overshooting region and found some properties of the overshooting region:
$\overline{u_{r}^{\prime}T^{\prime}}<0$, $\nabla_{R}<\nabla<\nabla_{ad}$, and
the peak of $\overline{T^{\prime}T^{\prime}}$, which are similar to
Xiong’s(1985) and Xiong & Deng’s(2001) works. In this section, we attempt to
get semi-analytical solutions of the TCM.

Some approximations are adopted to simplify Eqs.(1-5) in the overshooting
region:

Approximation I. Péclet number $P_{e}\gg 1$, where $P_{e}=\frac{\rho
C_{P}l\sqrt{k}}{\lambda}$. That is
$\frac{\varepsilon}{k}\gg\frac{\lambda}{\rho
c_{P}}\frac{\varepsilon^{2}}{k^{3}}$ which means the turbulent dissipation is
much stronger than the thermal dissipation. This assumption is reasonable in
most cases except for the region near the surface of a star or with very small
$k$.

Approximation II. All variations, except the turbulent fluctuations, are
thought to be constant because the turbulent fluctuations change much faster
than others in the overshooting region.

Approximation III. Far away from the convective boundary,
$\nabla\approx\nabla_{R}$. This assumption is acceptable if the heat flux
$\overline{u_{r}^{\prime}T^{\prime}}$ is small.

###  3.1 Turbulent heat transport in the overshooting region

Defining the anisotropic degree
$\omega=\frac{\overline{u_{r}^{\prime}u_{r}^{\prime}}}{2k}$ which is the ratio
of radial kinetic energy to total kinetic energy, and applying Approximation
II and Eq.(5), we can rewrite Eq.(3) to:

$\displaystyle\frac{\partial}{\partial r}\left(4C_{t1}\omega
l\sqrt{k}\frac{\partial\overline{u_{r}^{\prime}T^{\prime}}}{\partial
r}\right)=-\frac{T}{H_{P}}(\nabla_{R}-\nabla_{ad})\overline{u_{r}^{\prime}u_{r}^{\prime}}+\frac{\beta
g_{r}}{T}\overline{T^{\prime}T^{\prime}}+[2\omega
P_{e}+C_{t}(1+P_{e}^{-1})]\frac{\sqrt{k}}{l}\overline{u_{r}^{\prime}T^{\prime}}$
(6)

In the last bracket in Eq.(6), Approximation I($P_{e}\gg 1$) makes the
dissipation term
$C_{t}(1+P_{e}^{-1})\frac{\sqrt{k}}{l}\overline{u_{r}^{\prime}T^{\prime}}$ be
ignorable. And, by using Eq.(5) and Approximation II, it is easy to find that
the diffusion term is on the same order of the ignorable dissipation term:

$\displaystyle\frac{\partial}{\partial r}\left(4C_{t1}\omega
l\sqrt{k}\frac{\partial\overline{u_{r}^{\prime}T^{\prime}}}{\partial
r}\right)\approx
2C_{t1}\alpha^{2}\omega\frac{dlnk}{dlnP}\cdot\frac{dln(\nabla_{R}-\nabla)}{dlnP}(\frac{\sqrt{k}}{l}\overline{u_{r}^{\prime}T^{\prime}})\sim
Pe^{0}(\frac{\sqrt{k}}{l}\overline{u_{r}^{\prime}T^{\prime}})$ (7)

Therefore the diffusion term is also ignorable. Equation (3) is in local
equilibrium:

$\displaystyle-\frac{T}{H_{P}}(\nabla-\nabla_{ad})\overline{u_{r}^{\prime}u_{r}^{\prime}}+\frac{\beta
g_{r}}{T}\overline{T^{\prime}T^{\prime}}\approx 0$ (8)

In the overshooting region, the most important process is the diffusion of the
kinetic energy. Thus, we ignore the diffusion of
$\overline{T^{\prime}T^{\prime}}$(i.e., setting $C_{e1}=0$). The solution of
the TCM with $C_{e1}=0$ can be thought as the zero-order solution of the TCM.

Ignoring the diffusion of $\overline{T^{\prime}T^{\prime}}$ and the diffusion
and dissipation terms of $\overline{u_{r}^{\prime}T^{\prime}}$, using
Approximations I & II, one can rewrite Eqs.(1-4) as:

$\displaystyle\frac{2C_{s}l}{k}\frac{\partial}{\partial r}(\omega
k^{\frac{5}{2}}\frac{\partial\omega}{\partial
r})=(C_{k}-1)(\omega-\frac{1}{3})\frac{k^{\frac{3}{2}}}{l}+\frac{\beta
g_{r}}{T}\overline{u_{r}^{\prime}T^{\prime}}(1-\omega)$ (9) $\displaystyle
2C_{s}l\frac{\partial}{\partial r}(\omega k^{\frac{1}{2}}\frac{\partial
k}{\partial r})=\frac{k^{\frac{3}{2}}}{l}+\frac{\beta
g_{r}}{T}\overline{u_{r}^{\prime}T^{\prime}}$ (10) $\displaystyle
0=-\frac{2T}{H_{P}}(\nabla-\nabla_{ad})\omega k+\frac{\beta
g_{r}}{T}\overline{T^{\prime}T^{\prime}}$ (11) $\displaystyle
0=-\frac{2T}{H_{P}}(\nabla-\nabla_{ad})\overline{u_{r}^{\prime}T^{\prime}}+2C_{e}\frac{\varepsilon}{k}\overline{T^{\prime}T^{\prime}}$
(12)

Equation (9) results from Eq.(1) and (2), describing the equilibrium structure
of the anisotropic degree $\omega$. The left side is the diffusion of
$\omega$. The first term in the right side is the dissipation rate due to
return to isotropy term in Eq.(1). The last term is the generation rate of
$\omega$ due to the buoyancy.

Equations (11) and (12) show:

$\displaystyle
0=(\nabla-\nabla_{ad})(\overline{u_{r}^{\prime}T^{\prime}}+2C_{e}\omega\frac{T}{\beta
g}\frac{k^{\frac{3}{2}}}{l})$ (13)

The solution is
$\overline{u_{r}^{\prime}T^{\prime}}=-2C_{e}\varepsilon\omega\frac{T}{\beta
g}$ or $\nabla=\nabla_{ad}$. The latter is equivalent to
$\overline{u_{r}^{\prime}T^{\prime}}=-\frac{T}{H_{P}}\frac{\lambda}{\rho
c_{P}}(\nabla_{ad}-\nabla_{R})$. Because $\overline{u_{r}^{\prime}T^{\prime}}$
is close to zero near the convective boundary and gradually decreases far away
from the convective boundary(Xiong, 1989; Xiong & Deng, 2001; Zhang & Li,
2009), the physically acceptable result is:

$\displaystyle\overline{u_{r}^{\prime}T^{\prime}}=Max\\{-\frac{T}{H_{P}}\frac{\lambda}{\rho
c_{P}}(\nabla_{ad}-\nabla_{R}),-2C_{e}\omega\frac{T}{\beta
g}\frac{k^{\frac{3}{2}}}{l}\\}$ (14)

Equation (14) shows that there is an adiabatic stratification zone in the
overshooting region in the case of $C_{e1}=0$. In order to investigate the
property of heat transport in the overshooting region, we must know the length
of the adiabatic stratification zone. It is found in Eq.(14) that the boundary
of the adiabatic stratification is the location where
$\frac{T}{H_{P}}\frac{\lambda}{\rho
c_{P}}(\nabla_{ad}-\nabla_{R})=2C_{e}\omega\frac{T}{\beta
g}\frac{k^{\frac{3}{2}}}{l}$. Solving the equation of $\omega$ is not easy
because it is nonlinear. However, this problem is avoidable. Turbulent motions
are isotropic when $\omega=\frac{1}{3}$. In the convection zone,
$\omega>\frac{1}{3}$ because the buoyancy boosts radial turbulent motion. In
most part of overshooting region, $\omega$ should be less than $\frac{1}{3}$
because the buoyancy prevents radial turbulent motion. Therefore $\omega$
should be not far away from $\frac{1}{3}$ near the convective boundary.
Further more, taking $\omega$ as a constant, one can rewrite Eq.(10) as:

$\displaystyle 2C_{s}l\omega\frac{\partial}{\partial
r}(k^{\frac{1}{2}}\frac{\partial k}{\partial
r})=\frac{k^{\frac{3}{2}}}{l}+\frac{\beta
g_{r}}{T}\overline{u_{r}^{\prime}T^{\prime}}$ (15)

Substituting Eq.(14) into the above equation, one can get the approximate
solution:

$\displaystyle k^{\frac{3}{2}}\approx
k_{C}^{\frac{3}{2}}exp(-\sqrt{\frac{3}{4C_{s}\omega}}|\frac{r-r_{C}}{l}|)$
(16)

if $\frac{T}{H_{P}}\frac{\lambda}{\rho c_{P}}(\nabla_{ad}-\nabla_{R})\leq
2C_{e}\omega\frac{T}{\beta g}\frac{k^{\frac{3}{2}}}{l}$, and:

$\displaystyle
k^{\frac{3}{2}}=k_{A}^{\frac{3}{2}}exp(-\sqrt{\frac{3(1+2C_{e}\omega)}{4C_{s}\omega}}|\frac{r-r_{A}}{l}|)$
(17)

if $\frac{T}{H_{P}}\frac{\lambda}{\rho
c_{P}}(\nabla_{ad}-\nabla_{R})>2C_{e}\omega\frac{T}{\beta
g}\frac{k^{\frac{3}{2}}}{l}$.

In Eq.(16), point C, which is the convective boundary where
$\nabla_{ad}=\nabla_{R}$, is set to be the initial point, $k_{C}$ and $r_{C}$
being $k$ and $r$ here. The contribution of the buoyancy term(i.e. the last
term in Eq.(15)) is ignored in obtaining the solution Eq.(16). In the deep
convection zone, turbulent motions are almost in local equilibrium, thus the
ratio of $-\frac{\beta g_{r}}{T}\overline{u_{r}^{\prime}T^{\prime}}$ to
$\frac{k^{\frac{3}{2}}}{l}$ is about 1. However, near the convective boundary,
buoyancy is about zero, meanwhile the diffusion of $k$ dominates. Those make
the ratio be much less than 1. Therefore the buoyancy term is ignorable.

In Eq.(17), point A, where $k=k_{A}$ and $r=r_{A}$, is the boundary of the
adiabatic overshooting region. In the region beyond point A, the ratio of
$-\frac{\beta g_{r}}{T}\overline{u_{r}^{\prime}T^{\prime}}$ to
$\frac{k^{\frac{3}{2}}}{l}$ is $2C_{e}\omega$ which is on the order of $1$,
thus the buoyancy term remains.

The exponentially decreasing function of $k$ is due to the fact that there is
no generation in the overshooting region. Contrary to the situation in the
convection zone, the buoyancy dissipates $k$ because it prevents the radial
motion of fluid elements in the overshooting region. The distribution of $k$
results from the equilibrium between the diffusion and the dissipation. $k$
should decrease faster if the buoyancy is as effective as the turbulent
dissipation, which is found by comparing the exponential indices of Eq.(16)
and (17).

The location of point A is determined by $\frac{T}{H_{P}}\frac{\lambda}{\rho
c_{P}}(\nabla_{ad}-\nabla_{R})=2C_{e}\omega\frac{T}{\beta
g}\frac{k^{\frac{3}{2}}}{l}$. Using Eq.(16), we get a property of point A:

$\displaystyle
k_{C}^{\frac{3}{2}}exp(-\sqrt{\frac{3}{4C_{s}\omega}}|\frac{r_{A}-r_{C}}{l}|)=\frac{1}{2C_{e}\omega}\frac{\alpha\beta
g\lambda}{\rho c_{P}}(\nabla_{ad}-\nabla_{R,A})$ (18)

The relation between $r_{A}$ and $\nabla_{R,A}$ is needed in order to solve
this equation and to locate point A. Near the convective boundary, there is:

$\displaystyle|\nabla_{ad}-\nabla_{R,A}|\approx\nabla_{ad}|\chi(lnP_{A}-lnP_{C})|=\nabla_{ad}|\chi|\frac{l_{ad}}{H_{P}}$
(19)

where $l_{ad}=|r_{A}-r_{C}|$ is the length of the adiabatic overshooting
region, $P_{A}$ and $P_{C}$ the pressure at point A and C, and
$\chi=\frac{dln\nabla_{R}}{dlnP}$ which is approximately a constant.

Substituting Eq.(19) into Eq.(18), one finds:

$\displaystyle
k_{C}^{\frac{3}{2}}exp(-\frac{1}{\alpha}\sqrt{\frac{3}{4C_{s}\omega}}\frac{l_{ad}}{H_{P}})=\frac{1}{2C_{e}\omega}\frac{\alpha\beta
g\lambda}{\rho c_{P}}\nabla_{ad}|\chi|\frac{l_{ad}}{H_{P}}$ (20)

$l_{ad}$ can be worked out if $k_{C}$ is known. In the deep adiabatic
convection zone, turbulent diffusion is ignorable, and the localized TCM shows
$k^{\frac{3}{2}}_{Local}=\frac{\alpha\beta
g\lambda(\nabla_{R}-\nabla_{ad})}{\rho c_{P}}$ (see Appendix A). However,
$k_{C}$ can not be estimated as that because $\nabla_{R}=\nabla_{ad}$ thus
$k_{Local}=0$ at the convective boundary. Actually, the turbulent diffusion of
$k$ is effective near the convective boundary, and $k_{C}$ is determined by
the diffusion. We can estimate $k_{C}$ by a simple approach which will be
referred to as the maximum of diffusion hereafter. Setting point B at where
the diffusion becomes dominative in the convection zone, we get the relation
between $k_{C}$ and $k_{B}$ by solving Eq.(15):

$\displaystyle
k_{C}^{\frac{3}{2}}=k_{B}^{\frac{3}{2}}exp(-\sqrt{\frac{3}{4C_{s}\omega}}|\frac{r_{C}-r_{B}}{l}|)$
(21)

where $k_{B}^{\frac{3}{2}}\approx\frac{\alpha\beta
g\lambda(\nabla_{R,B}-\nabla_{ad})}{\rho c_{P}}$. Equation (21) shows that
$k_{C}$ is a function of $r_{B}$. In reality, the diffusion leads to the
maximum of $k_{C}$. Therefore $r_{B}$ makes the derivation of the right side
of Equation (21) be zero. Noting that $\nabla_{R,B}-\nabla_{ad}$ is
approximately proportional to $r_{B}-r_{C}$, one can easily work out the
location of point B:

$\displaystyle\sqrt{\frac{3}{4C_{s}\omega}}|\frac{r_{C}-r_{B}}{l}|\approx 1$
(22)

It is found in Fig.1 that $k\approx k_{Local}$ in the deep convection zone
because the turbulent diffusion can be ignored here, and the turbulent
diffusion dominates in the layer beyond point B.

Using above results, we obtain:

$\displaystyle k_{C}^{\frac{3}{2}}=\frac{1}{e}\frac{\alpha\beta
g\lambda(\nabla_{R,B}-\nabla_{ad})}{\rho
c_{P}}\approx\frac{1}{e}\sqrt{\frac{4C_{s}\omega}{3}}\frac{\alpha^{2}\beta
g\lambda\nabla_{ad}|\chi|}{\rho c_{P}}$ (23)

Generally, $\frac{l_{ad}}{H_{P}}$ is very small. According to Eq.(20), the
length of the adiabatic overshooting region is:

$\displaystyle
l_{ad}\approx\frac{\sqrt{\frac{4C_{s}\omega}{3}}}{\frac{e}{2C_{e}\omega}+1}l$
(24)

In the area $|r-r_{C}|\leq|r_{A}-r_{C}|$ in the overshooting region, the
temperature gradient $\nabla$ is almost equal to the adiabatic one. In the
area $|r-r_{C}|>|r_{A}-r_{C}|$, however, according to Eq.(14), Eq.(17), and
Eq.(5), the temperature gradient $\nabla$ is gradually close to $\nabla_{R}$:

$\displaystyle\nabla-\nabla_{R}=(\nabla_{ad}-\nabla_{R,A})\cdot
exp[-\sqrt{\frac{3(1+2C_{e}\omega)}{4C_{s}\omega}}|\frac{r-r_{A}}{l}|]$ (25)

Although $\omega$ in Eq.(24) and Eq.(25) is still unknown, we can estimate it
roughly. Equation (24) and (25) describe the turbulent motion near the
convective boundary, thus we can use $\omega\approx\omega_{C}$ where
$\omega_{C}$ is $\omega$ at the convective boundary. In the deep convection
zone, $\omega$ is almost equal to the equilibrium value
$\omega_{cz}=\frac{2}{3C_{k}}+\frac{1}{3}$ which is derived from the localized
TCM (see Appendix A). $\omega_{C}<\omega_{cz}$ because the buoyancy is zero at
the boundary, and $\omega_{C}>\frac{1}{3}$ because the diffusion of $\omega$.
Therefore the typical value of $\omega_{C}$ can be taken as the average, i.e.
$\omega_{C}\approx\frac{1}{2}(\omega_{cz}+\frac{1}{3})$. If Eq.(25) is used in
the region far away from the convective boundary(beyond the peak of
$\overline{T^{\prime}T^{\prime}}$), $\omega\approx\omega_{C}$ is not
appropriate. One can use $\omega=\omega_{o}$, where $\omega_{o}$ is the
equilibrium value of $\omega$ in the overshooting region which is introduced
in the next subsection.

Another turbulent correlation is $\overline{T^{\prime}T^{\prime}}$, which can
be worked out by using Eq.(11):

$\displaystyle\overline{T^{\prime}T^{\prime}}\approx
0,(|r-r_{C}|\leq|r_{A}-r_{C}|)$ (26)

And:

$\displaystyle\overline{T^{\prime}T^{\prime}}=\frac{2T}{H_{P}}\frac{T}{\beta
g}(\nabla_{ad}-\nabla)\omega k,(|r-r_{C}|>|r_{A}-r_{C}|)$ (27)

Equation (26) seems to against Cauchy’s theorem
$\overline{u_{r}^{\prime}u_{r}^{\prime}}\overline{T^{\prime}T^{\prime}}\geq\overline{u_{r}^{\prime}T^{\prime}}^{2}$.
Actually, ${\overline{T^{\prime}T^{\prime}}\approx 0}$ is only an approximate
solution on the order of
($Pe^{1}\frac{\sqrt{k}}{l}\overline{u_{r}^{\prime}T^{\prime}}$), because
Eq.(8) is an approximation on that order. Numerical calculations show no
confliction.

Results obtained above are based on $C_{e1}=0$. Numerical results of $\nabla$
with both $C_{e1}=0$ and $C_{e1}\neq 0$ are shown in Fig.2. It is found that
the effects of the diffusion of $\overline{T^{\prime}T^{\prime}}$ are only
making $\nabla$ be smoother. However, there is no adiabatic overshooting
region when the diffusion of $\overline{T^{\prime}T^{\prime}}$ is present,
because $\overline{T^{\prime}T^{\prime}}$ increases near the convective
boundary due to the turbulent diffusion thus $\nabla$ decreases according to
Eq.(8). Numerical results of the turbulent correlations in both $C_{e1}=0$ and
$C_{e1}\neq 0$ with different TCM parameters and for different stellar models
are shown in Figs.3-5. It is found that the theoretical solutions well fit the
numerical solutions in the case of $C_{e1}=0$. This also validates that the
boundary value $k_{C}$ derived from the maximum of diffusion is a good
approximation. The diffusion of $\overline{T^{\prime}T^{\prime}}$ modifies and
smoothes the profile of $\overline{T^{\prime}T^{\prime}}$ and
$\overline{u_{r}^{\prime}T^{\prime}}$. However, $k$ is insensitive to the
diffusion of $\overline{T^{\prime}T^{\prime}}$ because that $k$ is mainly
dominated by the diffusion of itself. The diffusion of
$\overline{T^{\prime}T^{\prime}}$ doesn’t significantly change the integral
value of $\overline{T^{\prime}T^{\prime}}$. According to Eq.(8), the integral
value of $\nabla$ or $\overline{u_{r}^{\prime}T^{\prime}}$ is also insensitive
to the diffusion of $\overline{T^{\prime}T^{\prime}}$, which is found in
Figs.(2-5).

The distribution of $\overline{T^{\prime}T^{\prime}}$ reveals an important
property of the overshooting. In the nonadiabatic overshooting region, using
$\nabla\approx\nabla_{R}$, one finds that
$\overline{T^{\prime}T^{\prime}}\propto T(\nabla_{ad}-\nabla_{R})k$ according
to Eq.(27). This result indicates a maximum of
$\overline{T^{\prime}T^{\prime}}$(Xiong, 1985; Zhang & Li, 2009) which is
shown in Figs.3-5. Beyond the location of the maximum of
$\overline{T^{\prime}T^{\prime}}$, the temperature of a turbulent element is
gradually close to the temperature of the environment, and the efficiency of
heat transport significantly decreases. Therefore the area between the
convective boundary and the location of the maximum of
$\overline{T^{\prime}T^{\prime}}$ can be thought as the overshooting region of
$\overline{u_{r}^{\prime}T^{\prime}}$. It is found in Figs.3-5 that the width
of the valley of $\overline{u_{r}^{\prime}T^{\prime}}$ is approximately equal
to the distance from the convective boundary to the location of the maximum of
$\overline{T^{\prime}T^{\prime}}$. In order to get the overshooting length of
heat transport, we need to locate the maximum of
$\overline{T^{\prime}T^{\prime}}$.

Using Eq.(17), defining
$\theta_{0}=\frac{dlnk}{dlnP}=\pm\frac{1}{\alpha}\sqrt{\frac{(1+2C_{e}\omega)}{3C_{s}\omega}}$
as the decaying index of $k$ (in the case of $C_{e1}=0$), we get:

$\displaystyle\overline{T^{\prime}T^{\prime}}\propto
T(\nabla_{ad}-\nabla_{R})P^{\theta_{0}}$ (28)

The derivative of $\overline{T^{\prime}T^{\prime}}$ is zero at the peak of
$\overline{T^{\prime}T^{\prime}}$. We get $\nabla_{R}$ there(denoted as
$\nabla_{R}^{*}$):

$\displaystyle(\nabla_{R}^{*}+\theta_{0})(\nabla_{ad}-\nabla_{R}^{*})-\chi\nabla_{R}^{*}\approx
0$ (29)

$\nabla_{R}^{*}$ is determined by only one turbulent parameter $\theta_{0}$.

The typical overshooting length of $\overline{u_{r}^{\prime}T^{\prime}}$ (or
$\nabla$) can be estimated with $\nabla_{R}^{*}$:

$\displaystyle|\chi|=|\frac{dln\nabla_{R}}{dlnP}|\approx\
|\frac{ln\nabla_{R,C}-ln\nabla_{R}^{*}}{lnP_{C}-lnP^{*}}|=|\frac{ln\nabla_{ad}-ln\nabla_{R}^{*}}{lnP_{C}-lnP^{*}}|=\frac{ln\frac{\nabla_{ad}}{\nabla_{R}^{*}}}{\frac{l_{\nabla}}{H_{P}}}$
(30)

where $\nabla_{R,C}$ is $\nabla_{R}$ at the convective boundary, $l_{\nabla}$
is the distance from the convective boundary to the location of the maximum of
$\overline{T^{\prime}T^{\prime}}$ and also the typical overshooting length of
$\nabla$.

$l_{\nabla}$ is worked out as:

$\displaystyle
l_{\nabla}\approx\frac{1}{|\chi|}ln\frac{\nabla_{ad}}{\nabla_{R}^{*}}H_{P}$
(31)

Usually, $|\theta_{0}|$ is much larger than $|\chi|$ and $\nabla_{ad}$, and
$\nabla_{R}^{*}$ can be approximately solved from Eq.(29):

$\displaystyle\nabla_{R}^{*}\approx(1-\frac{\chi}{\theta_{0}})\nabla_{ad}$
(32)

Finally, we find:

$\displaystyle l_{\nabla}\approx\frac{H_{P}}{|\theta_{0}|}=H_{k}$ (33)

where $H_{k}$ is the scale height of turbulent kinetic energy $k$ defined by
$H_{k}=|\frac{dr}{dlnk}|$. The result indicates that $\nabla$ is remarkably
modified by the overshooting only in about $1H_{k}$. It is found in Fig.3 that
$l_{\nabla}=ln\frac{k_{C}}{k_{*}}H_{k}\approx 0.8H_{k}$, which is in agreement
with Eq.(33). It is shown in Fig.2 that $\nabla$ is remarkably modified only
in $1H_{k}$.

###  3.2 Asymptotic analysis

In above subsection, we have discussed the turbulent heat transport and the
solution of turbulent correlations in the overshooting region near the
convective boundary based on the assumption $C_{e1}=0$. The diffusion of
$\overline{T^{\prime}T^{\prime}}$ only modifies turbulent correlations to be
smoother near the convective boundary. However, it makes more effects on
turbulent motions in the overshooting region further than $1H_{k}$ away from
the convective boundary. In this subsection, we investigate the turbulence
properties in the outer overshooting region(beyond $1H_{k}$).

In the numerical calculations of the TCM, we found that the anisotropic degree
$\omega$ always showed an equilibrium value in the overshooting region. A
typical numerical result is shown in Fig.6. In order to understand it, we
discuss the behave of the anisotropic degree $\omega$ in both convection zone
and overshooting region. $\omega$ should be larger than $\frac{1}{3}$ in the
convection zone because the buoyancy boosts radial movement of turbulent
elements. Actually, $\omega$ is almost equal to the equilibrium value in the
convection zone $\omega_{cz}=\frac{2}{3C_{k}}+\frac{1}{3}$(see Appendix A) due
to the equilibrium between the buoyancy and the return to isotropy term. When
turbulent elements go across the convective boundary into the overshooting
region, the buoyancy prevents convective elements moving, thus $\omega$
decreases to less than $\frac{1}{3}$ near the convective boundary. However, as
$\overline{u_{r}^{\prime}T^{\prime}}$ exponentially decreasing, the
equilibrium of $\omega$ is established again in the overshooting region. This
results in an asymptotic property of the overshooting region: there is an
equilibrium value of $\omega$ in the overshooting region,
$\omega\approx\omega_{o}$.

By using the asymptotic property $\omega\approx\omega_{o}$ and Approximations
I, II & III, it is easy to get the asymptotic solution of TCM in the
overshooting region(see Appendix B):

$\displaystyle\overline{u_{r}^{\prime}T^{\prime}}=\frac{(C_{k}-1)(\omega_{o}-\frac{1}{3})}{(1-\omega_{o})}\frac{T}{\beta
g}\frac{k^{\frac{3}{2}}}{l}$ (34)
$\displaystyle\overline{T^{\prime}T^{\prime}}=2\omega_{o}(\nabla_{ad}-\nabla_{R})\frac{T^{2}}{\beta
gH_{P}}k$ (35) $\displaystyle k=k_{0}(\frac{P}{P_{0}})^{\theta}$ (36)

where $\theta$ is the asymptotic solution of $\frac{dlnk}{dlnP}$:

$\displaystyle\theta=\pm\frac{1}{\alpha}\sqrt{\frac{1}{3C_{s}\omega_{o}}[1-\frac{(C_{k}-1)(\omega_{o}-\frac{1}{3})}{(1-\omega_{o})}]}$
(37)

$k$ takes the decreasing expression in the overshooting region, which means:
${}^{\prime}+^{\prime}$ is adopted in the upward overshooting region and
${}^{\prime}-^{\prime}$ in the downward one.

The equilibrium value $\omega_{o}$ is determined by:

$\displaystyle(2C_{s}C_{e}-C_{e1}C_{k}){\omega_{o}}^{2}+[\frac{1}{3}C_{e1}(C_{k}+2)-C_{s}(C_{k}+2C_{e}-1)]\omega_{o}+\frac{1}{3}C_{s}(C_{k}-1)=0$
(38)

The equilibrium value $\omega_{o}$ is only a function of turbulent parameters
$(C_{e},C_{e1},C_{s},C_{k})$. The fact that the buoyancy prevents the radial
movement of turbulent elements in the overshooting region restricts the
turbulent parameters to ensure $\omega_{o}<\frac{1}{3}$.

An important thing is where $\omega$ reaches its equilibrium value
$\omega_{o}$. According to Eq.(9), the equilibrium of $\omega$ can be realized
only if the buoyancy term synchronically decreases with $k$ decreasing.
Therefore $\omega$ starts to reach its equilibrium value $\omega_{o}$ beyond
the peak of $\overline{T^{\prime}T^{\prime}}$ due to
$|\overline{u_{r}^{\prime}T^{\prime}}|$ being decreasing.

Setting $C_{e1}=0$ in Eq.(38), we find that the asymptotic solution is the
same as the results in the overshooting region with
$|r-r_{C}|\geq|r_{A}-r_{C}|$ by setting $\omega=\omega_{o}$ in Eq(14),(17) &
(27). Because Eq.(8) is correct whether $C_{e1}=0$ or not, the conclusion that
the maximum of $\overline{T^{\prime}T^{\prime}}$ is located at about $1H_{k}$
is also correct in both cases.

It must be mentioned that we have used Approximation I(i.e. $P_{e}\gg 1$),
which means that the turbulent dissipation is much larger than the thermal
dissipation. If $k$ decreases enough to satisfy $P_{e}\ll 1$, the thermal
dissipation should become significant thus $\overline{T^{\prime}T^{\prime}}$
and the turbulent kinetic energy $k$ should rapidly decrease to zero. Then
$\omega$ also rapidly decreases as shown in Fig.6. In another word, turbulent
movement can hardly overshoot into the thermal dissipation zone where
$P_{e}\ll 1$.

According to discussions above, we can separate the overshooting region into
three parts as shown in Fig.7: the overshooting region of
$\overline{u_{r}^{\prime}T^{\prime}}$ or $\nabla$ with the length of about
$1H_{k}$, the turbulent dissipation region in which the asymptotic solution
holds, and the thermal dissipation region in which the turbulent movement
quickly vanishes. The boundaries among those parts are the peak of
$\overline{T^{\prime}T^{\prime}}$ and the location of $P_{e}=1$.

##  4 Conclusions and discussions

Turbulent convection models are better tools in dealing with the convective
overshooting than non-local mixing length theories. However, they are often
too complex to be applied in the calculations of stellar structure and
evolution. In order to investigate the property of the convective overshooting
and to make it easy to apply turbulent convection models, we have analyzed the
TCM developed by Li & Yang (Li & Yang, 2007) and obtained approximate and
asymptotic solutions of the TCM in the overshooting region with $P_{e}\gg 1$.
The main conclusions and corresponding discussions are listed as follows:

1\. The overshooting region can be partitioned into three parts: a thin
turbulent heat flux overshooting region, a power law dissipation region of
turbulent kinetic energy, and a thermal dissipation area with rapidly
decreasing $k$. The turbulent fluctuations $k$,
$\overline{u_{r}^{\prime}T^{\prime}}$, and $\overline{T^{\prime}T^{\prime}}$
exponentially decrease in the overshooting region as Eqs.(34-36). The
equilibrium value of the anisotropic degree $\omega_{o}$ and the exponential
indices of the turbulent fluctuations are only determined by the parameters of
the TCM. The decaying behaviors of the turbulent fluctuations are similar to
Xiong & Deng’s results(Xiong, 1989; Xiong & Deng, 2001).

2\. The peak of $\overline{T^{\prime}T^{\prime}}$ in the overshooting region
is located at about $1H_{k}$ away from the convective boundary. In this
distance, the modification of $\nabla$ caused by the overshooting is
remarkable. An approximate profile of $\nabla$ comprises an adiabatic
overshooting region with the length of $l_{ad}$ and an exponentially
decreasing function, as described in Eq.(24) and (25). Beyond $1H_{k}$, the
modification of $\nabla$ is ignorable and $\nabla\approx\nabla_{R}$. It should
be noted that the result of $1H_{k}$ overshooting distance of turbulent heat
transfer is independent of the parameters of TCM, so it may be a general
property of the overshooting. Our result is similar to Marik & Petrovay(2002)
whose result shows that the length between the peak of
$\overline{T^{\prime}T^{\prime}}$ and the convective boundary is about
$1.2H_{k}$. Meakin & Arnett(2010) simulated the turbulent convection of a
$23M_{\odot}$ star, the data of the turbulent kinetic energy and the
convective flux in the overshooting region being shown in Fig.8. It is found
that the overshooting length of the convective flux
$\overline{u_{r}^{\prime}T^{\prime}}$ is about $0.5\sim 2H_{k}$ which is in
agreement with our result.

3\. The value of the turbulent kinetic energy at the convective boundary
$k_{C}$ can be estimated by a method called the maximum of diffusion. The
value of turbulent fluctuations in the overshooting region can be estimated by
using the exponentially decreasing functions and the initial value $k_{C}$.
This may significantly simplify the application of the TCM in calculations of
the stellar structure and evolution.

There is a distinction between the non-local model of Zahn(1991) and our
results, i.e. the temperature gradient jumps from nearly adiabatic to
radiative in Zahn’s model but continuously changes in our results (see Fig.2).
This is caused by the assumption in Zahn’s model that the turbulent velocity
and temperature fluctuation are strongly correlated(Xiong & Deng, 2001). In
our results, the correlativity of turbulent velocity and temperature
fluctuation $R_{VT}=\frac{\overline{u_{r}^{\prime}T^{\prime}}}{\sqrt{2\omega
k\overline{T^{\prime}T^{\prime}}}}$ quickly decreases to zero then turns to be
negative near the convective boundary(see Fig.9), and the asymptotic solution
shows that $R_{VT}\propto\sqrt{k}$ and exponentially decreases in the
turbulent dissipation overshooting region. Our result is in agreement with
three-dimension simulations such as Fig.6 in Singh et al.(1995) and Fig.15 in
Meakin & Arnett(2007).

We thank the anonymous referee for valuable comments which help to improve the
paper. And we thank C. A. Meakin for providing the numerical data of Fig.8.
Fruitful discussions with J. Su, X. J. Lai and C. Y. Ding are highly
appreciated. This work is co-sponsored by the National Natural Science
Foundation of China through grant No.10673030 and No.10973035 and Science
Foundation of Yunnan Observatory No.Y0ZX011009.

##  Appendix A The localized TCM in convection zone.

The localized TCM results from Eqs.(1-4) by ignoring the diffusion terms. It
is a good approximate of the TCM in the convection zone(Li & Yang, 2001). We
attempt to work out the solution in this appendix.

Some symbols are defined for conveniences:
$U=\overline{u_{r}^{\prime}T^{\prime}}$, $V=\overline{T^{\prime}T^{\prime}}$,
$W=\sqrt{k}$, $A=\frac{T}{H_{P}}(\nabla_{R}-\nabla_{ad})$, $B=-\frac{\beta
g_{r}}{T}$, $D=\frac{\lambda}{\rho C_{P}}$,
$f=\frac{\nabla-\nabla_{ad}}{\nabla_{R}-\nabla_{ad}}$.

Ignoring the diffusion terms of Eqs.(1-4), we get the localized TCM:

$\displaystyle
0=\frac{2}{3}\frac{W^{3}}{l}-2BU+2C_{k}(\omega-\frac{1}{3})\frac{W^{3}}{l}$
(A1) $\displaystyle 0=\frac{W^{3}}{l}-BU$ (A2) $\displaystyle 0=-2\omega
fAW^{2}-BV+C_{t}(1+P_{e}^{-1})\frac{WU}{l}$ (A3) $\displaystyle
0=-2fAU+2C_{e}(1+P_{e}^{-1})\frac{WV}{l}$ (A4) $\displaystyle U=AD(1-f)$ (A5)

Equation (A1) and (A2) show:

$\displaystyle\omega=\frac{2}{3C_{k}}+\frac{1}{3}$ (A6)

This is the equilibrium value $\omega_{cz}$ in convection zone.

Describing $W$, $V$, $U$ by $f$ and $P_{e}$($=\frac{lW}{D}$), we find:

$\displaystyle
f=\frac{C_{t}C_{e}P_{e}^{-1}(1+P_{e}^{-1})^{2}}{C_{t}C_{e}P_{e}^{-1}(1+P_{e}^{-1})^{2}+2C_{e}\omega(1+P_{e}^{-1})+1}$
(A7)

$W$, $V$ can be worked out as:

$\displaystyle W^{3}=ABDl(1-f)$ (A8) $\displaystyle
V=\frac{AfW^{2}}{C_{e}B(1+P_{e}^{-1})}$ (A9)

According to $P_{e}=\frac{lW}{D}$, Eq.(A8) and Eq.(A7), we get the equation of
$P_{e}$:

$\displaystyle aP_{e}^{4}+(b+1)P_{e}^{3}+2bP_{e}^{2}+(b-at)P_{e}-t=0$ (A10)

where $a=1+\frac{1}{2\omega C_{e}}$, $b=\frac{C_{t}}{2\omega}$,
$t=\frac{ABl^{4}}{D^{2}}$. $f$ is determined by $f=1-\frac{P_{e}^{3}}{t}$
according to Eq.(A8).

Solving Eq.(A10), we can obtain all turbulent fluctuations of the localized
TCM by using Eq.(A5), (A8), (A9) and (A11).

An important case is $t\gg 1$, thus $P_{e}\gg 1$ according to Eq.(A10). In
that case, Eq.(A7) shows:

$\displaystyle f=\frac{C_{e}C_{t}P_{e}^{-1}}{2C_{e}\omega+1}\approx 0$ (A11)

which corresponds to the adiabatic convection.

Finally, we obtain the turbulent fluctuations according to Eq.(A8), (A5) &
(A9):

$\displaystyle W^{3}\approx ABDl$ (A12) $\displaystyle
V\approx\frac{C_{t}}{2C_{e}\omega+1}\frac{AD}{Bl}W$ (A13) $\displaystyle
U\approx AD$ (A14)

and the correlativity of turbulent velocity and temperature $R_{VT}$:

$\displaystyle R_{VT}=\frac{U}{\sqrt{2\omega
W^{2}V}}\approx\sqrt{\frac{2C_{e}\omega+1}{2C_{t}\omega}}$ (A15)

##  Appendix B Details of deriving the asymptotic solution of the TCM in
overshooting region.

There are the details of obtaining the asymptotic solution of the TCM in
overshooting region.

Some symbols are defined for conveniences:
$U=\overline{u_{r}^{\prime}T^{\prime}}$, $V=\overline{T^{\prime}T^{\prime}}$,
$W=\sqrt{k}$,
$A=-\frac{T}{H_{P}}(\nabla-\nabla_{ad})\approx-\frac{T}{H_{P}}(\nabla_{R}-\nabla_{ad})$
(Approximation III is used), $B=-\frac{\beta g_{r}}{T}$.

Applying the asymptotic property $\omega=\omega_{o}$ and Approximations I, II
& III, one can rewrite TCM as:

$\displaystyle
0=(C_{k}-1)(\omega_{o}-\frac{1}{3})\frac{W^{3}}{l}-BU(1-\omega_{o})$ (B1)
$\displaystyle lC_{s}\omega_{o}\frac{\partial}{\partial r}(W\frac{\partial
W^{2}}{\partial r})=\frac{W^{3}}{l}-BU$ (B2) $\displaystyle
0=-BV+2A\omega_{o}W^{2}$ (B3) $\displaystyle
lC_{e1}\omega_{o}\frac{\partial}{\partial r}(W\frac{\partial V}{\partial
r})=AU+\frac{C_{e}}{l}WV$ (B4)

Equation (B1) is equivalent to:

$\displaystyle
U=\frac{(C_{k}-1)(\omega_{o}-\frac{1}{3})}{(1-\omega_{o})}\frac{W^{3}}{Bl}$
(B5)

Taking it into Eq.(B2), one gets the equation of $W$:

$\displaystyle\frac{\partial^{2}W^{3}}{\partial
r^{2}}=\frac{3}{4C_{s}\omega_{o}l^{2}}[1-\frac{(C_{k}-1)(\omega_{o}-\frac{1}{3})}{(1-\omega_{o})}]W^{3}$
(B6)

Equation (B3) is equivalent to:

$\displaystyle V=\frac{2A\omega_{o}}{B}W^{2}$ (B7)

According to Eq.(B4), (B5) and (B7), one gets another equation of $W$:

$\displaystyle\frac{\partial^{2}W^{3}}{\partial
r^{2}}=\frac{3}{4C_{e1}{\omega_{o}}^{2}l^{2}}[\frac{(C_{k}-1)(\omega_{o}-\frac{1}{3})}{(1-\omega_{o})}+2C_{e}\omega_{o}]W^{3}$
(B8)

Comparing Eq.(B6) with Eq.(B8), one finds:

$\displaystyle\frac{3}{4C_{e1}{\omega_{o}}^{2}l^{2}}[\frac{(C_{k}-1)(\omega_{o}-\frac{1}{3})}{(1-\omega_{o})}+2C_{e}\omega_{o}]=\frac{3}{4C_{s}\omega_{o}l^{2}}[1-\frac{(C_{k}-1)(\omega_{o}-\frac{1}{3})}{(1-\omega_{o})}]$
(B9)

Therefore the equation of $\omega_{o}$ is:

$\displaystyle(2C_{s}C_{e}-C_{e1}C_{k}){\omega_{o}}^{2}+[\frac{1}{3}C_{e1}(C_{k}+2)-C_{s}(C_{k}+2C_{e}-1)]\omega_{o}+\frac{1}{3}C_{s}(C_{k}-1)=0$
(B10)

The asymptotic solution of $W$ is derived from Eq.(B6):

$\displaystyle
W=W_{0}exp{\\{\pm\frac{1}{2\alpha}\sqrt{\frac{1}{3C_{s}\omega_{o}}[1-\frac{(C_{k}-1)(\omega_{o}-\frac{1}{3})}{(1-\omega_{o})}]}ln(\frac{P}{P_{0}})\\}}$
(B11)

$W$ takes the decreasing expression in the overshooting region:
${}^{\prime}+^{\prime}$ is adopted in the upward overshooting region and
${}^{\prime}-^{\prime}$ in the downward one.

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Figure 1: Numerical results of $W=\sqrt{k}$, and
$W_{Local}\approx\sqrt[3]{\frac{\alpha\beta
g\lambda(\nabla_{R}-\nabla_{ad})}{\rho c_{P}}}$ which is the solution of
localized TCM (See Appendix A), for the solar model at present age. TCM
parameters are: $\alpha=0.84$, $C_{k}=2.5$, $C_{s}=0.1$, $C_{e1}=0$,
$C_{e}=0.2$, $C_{t}=7.0$, and $C_{t1}=0.01$, but $C_{t}$, $C_{t1}$ and
$C_{e1}$ are insensitive to the results. Point C is the boundary of the
convection zone, the location of point B is calculated by using Eq.(22).
Figure 2: Numerical results of temperature gradient near the convective
boundary in both $C_{e1}=0$ and $C_{e1}\neq 0$, $\nabla_{0}$ being the
temperature gradient of the model with $C_{e1}=0$, and $\nabla_{1}$
corresponding to $C_{e1}=0.01$. Dotted line $\nabla_{T}$, which is almost
identical to $\nabla_{0}$, is theoretical solution of the temperature gradient
with $C_{e1}=0$. The stellar model and other TCM parameters are the same as
Fig.1. Point A is the boundary of the adiabatic overshooting region. Our
theoretical result shows $l_{ad}\approx 0.013H_{P}$ in those TCM parameters
set, the numerical calculation being $0.015H_{P}$. Figure 3: Numerical results
of $\overline{T^{\prime}T^{\prime}}$, $\overline{u_{r}^{\prime}T^{\prime}}$,
$k$ near the convective boundary in both $C_{e1}=0$ and $C_{e1}\neq 0$, where
$U=\overline{u_{r}^{\prime}T^{\prime}}$, $W=\sqrt{k}$,
$V=\overline{T^{\prime}T^{\prime}}$. Dashed lines correspond to $C_{e1}=0$,
solid lines to $C_{e1}=0.01$. Dotted lines are the theoretical solutions with
$C_{e1}=0$. The stellar model and other TCM parameters are the same as Fig.1.
Figure 4: Numerical results of $\overline{T^{\prime}T^{\prime}}$,
$\overline{u_{r}^{\prime}T^{\prime}}$, $k$ near the convective boundary in
both $C_{e1}=0$ and $C_{e1}\neq 0$, where
$U=\overline{u_{r}^{\prime}T^{\prime}}$, $W=\sqrt{k}$,
$V=\overline{T^{\prime}T^{\prime}}$. Dashed lines correspond to $C_{e1}=0$,
solid lines to $C_{e1}=0.01$. Dotted lines are the theoretical solutions with
$C_{e1}=0$. The stellar model is a $7M_{\odot}$ star model at the top of RGB
phase. Others TCM parameters are: $\alpha=1.0$, $C_{k}=2.2$, $C_{s}=0.1$,
$C_{e}=1.0$, and $C_{t}=4.0$, $C_{t1}=0.01$. Figure 5: Numerical results of
$\overline{T^{\prime}T^{\prime}}$, $\overline{u_{r}^{\prime}T^{\prime}}$, $k$
near the boundary of the convective core in both $C_{e1}=0$ and $C_{e1}\neq
0$, where $U=\overline{u_{r}^{\prime}T^{\prime}}$, $W=\sqrt{k}$,
$V=\overline{T^{\prime}T^{\prime}}$. Dashed lines correspond to $C_{e1}=0$,
solid lines to $C_{e1}=0.01$. Dotted lines are the theoretical solutions with
$C_{e1}=0$. The stellar model is an early main sequence model of a
$3M_{\odot}$ star. Others TCM parameters are: $\alpha=1.0$, $C_{k}=2.1$,
$C_{s}=0.2$, $C_{e}=0.5$, and $C_{t}=3.0$, $C_{t1}=0.01$. Figure 6: Numerical
result of the structure of $\omega$ in overshooting region. The stellar model
is the solar model at present age. $C_{e1}=0.01$. The others TCM parameters
are the same as Fig.1, except $\alpha=0.2$ in order to enlarge $\theta$ to
show the thermal dissipation region in which $P_{e}\ll 1$. With those
parameters, the equilibrium value in convection zone is $\omega_{cz}=0.6$, and
the equilibrium value in overshooting region is $\omega_{o}=0.293$ which
denoted as the dotted line. Figure 7: The structure of the overshooting
region. $K=k$, $U=\overline{u_{r}^{\prime}T^{\prime}}$,
$V=\overline{T^{\prime}T^{\prime}}$. The stellar model is the solar model at
present age. $C_{e1}=0.01$, the others TCM parameters are the same as Fig.1,
except $\alpha=0.2$. With those parameters, in the turbulent dissipation
region with $P_{e}\gg 1$, theoretical result shows $\theta=17.5$ vs the
numerical result $17.6$, theoretical result of exponential decreasing index of
$U^{2}$ being $26.3$ vs the numerical result about $25.6$. $K$ is almost
parallel to $V$, which is in consistent with the asymptotic solution. In the
thermal dissipation region with $P_{e}\ll 1$, turbulent motion vanishes.
Figure 8: Numerical data of Meakin & Arnett (2010)’s results. The data of
model ’h1’ in their paper are plotted, where $U=F_{C}=\rho
C_{P}\overline{u_{r}^{\prime}T^{\prime}}$. Only the downward overshooting
region is shown. The distance from the convective boundary (where
$\overline{u_{r}^{\prime}T^{\prime}}=0$, about $R=0.62\times 10^{9}cm$) to the
right part of the valley of $\overline{u_{r}^{\prime}T^{\prime}}$ is about
$0.5\sim 2H_{k}$. Figure 9: Numerical results of the correlativity of
turbulent velocity and temperature $R_{VT}$. The stellar model is the solar
model at present age. Other TCM parameters are the same as Fig.1. $R_{VT}$
rapidly decreases to zero in the overshooting region. In the convection zone
near the convective boundary, the diffusion significantly enlarges
$\overline{T^{\prime}T^{\prime}}$ when $C_{e1}\neq 0$ (see Fig.3), and then
$R_{VT}$ is very small. In the interior of convection zone, localized TCM
shows the equilibrium value of $R_{VT}$ is
$R_{VT,cz}=\sqrt{\frac{2\omega_{cz}C_{e}+1}{2\omega_{cz}C_{t}}}$ (see Appendix
A). The TCM parameters show $R_{VT,cz}=0.384$.