A k- model for turbulently thermal convection in solar like and RGB stars

Li Yan

Yunnan Astronomical Observatory, CAS

1. Turbulently thermal convection in stars

2. Basic equations and the mixing length theory

3. k- model for stellar turbulent convection

4. Local solution of the k- model

5. Applications to the solar model

6. Applications to RGB stars

1. Turbulently thermal convection in stars

Thermal convection： In a gravitationally stratified fluid, the temperature gradient results in buoyancy to drive hot fluid moving upward and cold fluid moving downward. The structure of thermal convection is characterized by rolling cells, acting as thermal engines to transform heat into kinetic energy.

1. Turbulently thermal convection in stars

Effects of turbulently thermal convection in stars：

transfer of heat mixing of materials

Equation of mass continuity：

Navier-Stokes equations：

Equation of energy conservation：

0~~

~

ll

Uxt

l

ili

iil

li x

gx

PUU

xU

t

~

~~

~~~~~

~~~

~~~~~~

lll

l x

T

xUs

xs

tT

2. Basic equations and the mixing length theory

Mixing length theory： Convection cells move in average a mixing length l.

Equation of momentum：

Equation of energy conservation：

Equation of heat flux：

P

bP T

lH

gTTg

l

w

z

ww

1

,2

bPPP

badP

adb

lH

T

cz

T

cH

Tw

z

T

z

Tw

2

2

32

242 bP

PPPr H

glcTw

T

Hc

2. Basic equations and the mixing length theory

Equation of mixing length theory：Defining a heat transfer efficiency of convection：

we obtain the famous cubic equation of the MLT：

where

Question: What is the effect of rolling cell’s structure

How to determine the mixing length l

Solution is unavailable in stably stratified region

fxxxf 19

832

adr

adf

adrP

P

H

gTlc

x

2

e

1P

2. Basic equations and the mixing length theory

Equation of turbulent kinetic energy：

Equation of dissipation rate of turbulent kinetic energy：

where ,

Shear production rate：

Buoyancy production rate：

3. k- model for stellar turbulent convection

GPx

kkc

xDt

Dk

kk

2

k

ck

GcPcx

kc

xDt

D

kk

2

231

2

kkuuk2

1

m

l

m

l

x

u

x

u

kkugG

l

klk x

UuuP

Stellar structure:

Shear of convective rolling cells:

Temperature difference of convective rolling cells:

z

s

c

TgNc

P

g

z

sgg

PPi

2ad , ), 0, ,0(

r

rS

ln

)(

3. k- model for stellar turbulent convection

i

P

eg

VNT

r

VT

rr

Tr

rrc

2

2

2

2

11

The general solution is:

The size of a rolling cell is:

Averaged shear of rolling cells and shear production rate:

3. k- model for stellar turbulent convection

i

bb

b eR

rI

R

r

g

RNiT

)(1

2

Pb cR

22

222

bRk

Lk

S

k

Lcc

cS

kcP

P2

0

22

For the solar model：

typical length times typical velocity T times typical velocity

3. k- model for stellar turbulent convection

Model of convective heat flux：

Buoyancy frequency：

Buoyancy production rate：

3. k- model for stellar turbulent convection

22

22022

11

Nk

Nc

c

kcc

cG

PL

22

32

22

2

11

Nkkc

c

kcc

kcc

g

cF P

L

PL

LPC

adrPP

LP

L

PL

H

gT

kccc

kcc

kcc

N

2/522

22

22

2

1

1

In fully equilibrium state, the k- model reduces to:

where ,

Fully equilibrium state appears in the unstably stratified

region when c’ > 1, and in the stably stratified region when

c’ < 0

33311

1222

33

3

311

31222

1

13

1

11

13

4

ccccc

ccNc

cc

c

ccc

cccSc

331

123 3

111

cc

c

ccc

4. Local solution of the k- model

2 ,2

321 cc

In order to ensure positive shear production, we choose：

Fully equilibrium condition for shear production results in:

if

Turbulent kinetic energy should be proportional to the heat carried by convective rolling cells:

12

2200

3

111

kcccccc

PL

2

322

k

cL L

4. Local solution of the k- model

kc

c PL 2

bP

P Hg

NcTck

2

For convective cells moving adiabatically, we may assume:

In fully equilibrium state, it can be derived that:

It shows that the macro-length of turbulence

is proportional to local pressure scale height.

For convective cells in general, we assume:

LH b

PP

L Hg

NccL

222

4. Local solution of the k- model

kc

HR

g

cH

Pad

Pb

Pb

12

In fully equilibrium state, it can be modeled as:

We use this macro-length model of turbulence not only in the convection zone but also in the overshooting region.

In fully equilibrium state, we obtain:

where ,

4. Local solution of the k- model

4/1

2

2/12/3

2 1

kcc

HL

LPad

P

2

1

200

02/53/7 1

11

11x

c

yccc

cycyy

adrPad

PPe H

gTHc

xP

21

2

2 kccy PL

Compared with the MLT, the local solution of the k- model show similar asymptotic behavior in the limiting cases.

4. Local solution of the k- model

The sound speed of turbulent solar model is almost identical below the convection zone and higher in the convection zone than that of the MLT solar model.

5. Applications to the solar model

Comparisons of the typical velocity and typical length scale between the k- model and MLT.

5. Applications to the solar model

Comparisons of local and general solutions of the k- model

5. Applications to the solar model

Comparison of turbulent diffusivity resulted from the general solution of the k- model and from the MLT.

5. Applications to the solar model

For stars with different masses, the k- model results in bluer RGB sequences than the MLT does.

6. Applications to RGB stars

Comparisons of turbulent velocity and typical length for 1M

⊙ star in RGB bump stage

6. Applications to RGB stars

Comparisons of temperature gradient and turbulent diffusivity for 1M⊙ star in RGB bump stage

6. Applications to RGB stars

Comparisons of turbulent velocity and typical length for 3M

⊙ star at the top of RGB stage

6. Applications to RGB stars

Comparisons of temperature gradient and turbulent diffusivity for 3M⊙ star at the top of RGB stage

6. Applications to RGB stars

THANKS ！