Date post: | 21-Dec-2015 |
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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
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