Energy transportastro.phys.au.dk/jcd/astrofysik2/Lecture04_convection_handouts.pdf06/09/2013 1 Energy transport • Radiation (photons) • Convection (gas cells) • Heat transport

06/09/2013

1

Energy transport

• Radiation (photons)• Convection (gas

cells)• Heat transport

(between atoms and electrons)

• Particle radiation (neutrinos)

Convection

ConvectionSolar convection Rayleigh-Bénard convection

The instability condition

06/09/2013

2

ggfbuoy*

2rGmg

*

(6.1)The instability condition

0buoyf

2rGmg

*

Acceleration: Instable

0

0

gfbuoy


*1

*1 P

*2

*2 P

11 P

22 P

r


*1

*1 P

*2

*2 P

11 P

22 P

r

1*

1 PP 1

*1


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3

The motion is so slow that there is pressure balancebetween the element and the surroundings

How to determine Δρ?

min 3012/13

dyn

GMRt (6.2)

The motion is so slow that there is pressure balancebetween the element and the surroundings


min 3012/13

dyn

GMRt

How to determine Δρ?The motion is so slow that there is pressure balancebetween the element and the surroundings

The motion is so fast that there is no heat lossto the surroundings

min 3012/13

dyn

GMRt

yr mill. 30S

tot

S

2

KH L

URL

GMt (6.3)


The motion is so fast that there is no heat loss to the surroundings

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4


ggfbuoy*


2*2

but:

2*

2 PP


The motion is so fastthat there is no heat loss to the surroundings

2*

2 PP 2*

2 PP

Adiabatic


The motion is so fastthat there is no heat loss to the surroundings

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5

S

S

S

T

TP

P

ln ln1

ln ln

1

ln ln

3

2

2

1

Adiabatic exponents

entropy

S

S

S

T

TP

P

ln ln1

ln ln

1

ln ln

3

2

2

1

Adiabatic exponents

entropy

SS

PP

P

ln

ln 1

rrP

P

PP

PP

ddd11d

d1d1d

1

**

1*

*

1*

*

(6.4)

entropy

inside element outside


2*2


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6

2*2 12

*1

*2

(6.5)How to determine Δρ?

2*2

rr

rrP

P

dd

dd11

111

12*1

*2

rrP

P

PP

PP

ddd11d

d1d1d

1

**

1*

*

1*

*

(6.5)How to determine Δρ?

Taylor expansionof 2 about 1=1

*

2*2

rr

rrP

P

dd

dd11

111

12*1

*2

rrr

rrr

PP

dd

dd:

dd

dd1

ad

11

1

S

P

ln

ln 1


rrr

dd

dd

ad


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7

rrr

dd

dd

ad

0Instability if:


rrr

dd

dd

ad

0

rr dd

dd

ad

(6.7)


Instability if:

rr dd

dd

ad

rr dd

dd

ad

1

dd

rPP

1

dd

rPP

Negative

rr dd

dd

ad

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8

PP lndln d

lndln d1

ad1

If this condition is satisfied convection

(6.8)

rr dd

dd

ad

1

dd

rPP

1

dd

rPP

rr dd

dd

ad

PP lndln d

lndln d1

ad1

Convection when

”light”

”heavy”

rr dd

dd

ad

PP lndln d

lndln d1

ad1

”light”

”heavy”

53

Convection when Express instability condition in terms of T gradient

addd

dd

rT

rT

PP lndln d

lndln d

ad

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9

TP

km

B

u

rT

TrP

Pr dd1

dd1

dd1

(6.10)

Express instability condition in terms of T gradient

ideal EOS:TP

km

B

u

rT

TrP

Pr dd

dd

dd

rr dd

dd

ad

rT

TrP

PrP

P dd

dd

dd1

1

”definition”


ideal EOS:

TP

km

B

u

rT

TrP

Pr dd

dd

dd

rr dd

dd

ad

rT

TrP

PrP

P dd

dd

dd1

1

”definition”: i.e. (d/dr)ad is density gradient resulting from adiabatic motion in the given pressure gradient.


ideal EOS:

rr dd

dd

ad

rT

TrP

PrP

P dd

dd

dd1

1

rP

PrT

T dd1

dd

1

1

(6.11)

Correct thermodynamical treatment, which includes partial ionizationand departure from ideal gas law, shows that 1 in above equation mustbe replaced by 2.


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10

rr dd

dd

ad

rT

TrP

PrP

P dd

dd

dd1

1

rP

PrT

T dd1

dd

2

2

Correct thermodynamical treatment, which includes partial ionizationand departure from ideal gas law, shows that 1 in above equation mustbe replaced by 2.


rr dd

dd

ad

rP

PrT

T dd1

dd

2

2

ad

2

2

dd

dd

dd1

dd

rT

rT

T

rP

PT

rT

T

ad

2

2 : ln

ln1

SS PT

TP

PT

Adiabatic


rr dd

dd

ad

rP

PrT

T dd1

dd

2

2

ad

2

2

dd

dd

dd1

dd

rT

rT

T

rP

PT

rT

T

0

0

addd

dd

rT

rT

Instability when:

(6.12)


rr dd

dd

ad

rP

PrT

T dd1

dd

2

2

0

addd

dd

rT

rT

rr dd

dd

ad

Instability when:


ad

2

2

dd

dd

dd1

dd

rT

rT

T

rP

PT

rT

T

0

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11

rr dd

dd

ad

rP

PrT

T dd1

dd

2

2

0

addd

dd

rT

rT

Instability when:


ad

2

2

dd

dd

dd1

dd

rT

rT

T

rP

PT

rT

T

0

rr dd

dd

ad

rP

PrT

T dd1

dd

2

2

ad

2

2

dd

dd

dd1

dd

rT

rT

T

rP

PT

rT

T

0

0

addd

dd

rT

rT 1

dd

rP

TP

1

dd

rP

TP

Instability when:

Instability condition in terms of dimensionless T gradient

addd

dd

rT

rT 1

dd

rP

TP

1

dd

rP

TP

2

2 1 ln dln d

PT

(6.14)

Instability when:Instability condition in terms of dimensionless T gradient

This equation shows that instability sets in if the temperaturedecreases too rapidly outwards through the star.

addd

dd

rT

rT 1

dd

rP

TP

1

dd

rP

TP

2

2 1 ln dln d

PT

adadln

ln

PT

ad52

(6.16)

Instability when:

Convection sets in if

for fully ionized ideal gas

Instability condition in terms of dimensionless T gradient

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12

Where does convection occur?

Radiative energy transport

Energy transport by convection

32π16)( 3

dd

TrcarL

rT

Energy transport by radiation(dimensionless temperature gradient )

(6.17)

(5.8)

R

32π16)( 3

dd

TrcarL

rT

2

)( dd

rrGm

rP

Energy transport by radiation(dimensionless temperature gradient )R

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13

)( π16)( 3

dd

dd

2

32

1

rGmr

TrcarL

rT

rP


2

)( dd

rrGm

rP

)( π16)( 3

dd

dd

2

32

1

rGmr

TrcarL

rT

rP

XX X

X


2

)( dd

rrGm

rP

)(1

π16)( 3

dd

dd 3

1

rGmTcarL

rT

rP

umk

TP

B


umk

rGmTcarL

rT

rP

TP

)(1

π16)( 3

dd

dd B

3

1


umk

TP

B

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14

3B

)()(

π16 3

lndlnd

TrmrL

mGcak

PT

u


3B

R

)()(

π16 3

TrmrL

mGcak

u

(6.18)


3B

R

)()(

π16 3

TrmrL

mGcak

u

adR Instability when

Stability when adR

(6.19)

Criteria after Karl Schwarzschild (1905)


Where does convection occur?

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15

ad3B

)()(

π16 3

TrmrL

mGcak

u

Energy transport by convection(where does it occur?)

adR Instability when

Stability when adR

(6.19)

Criteria after Karl Schwarzschild (1905)

ad3B

)()(

π16 3

TrmrL

mGcak

u

If L/m is large. This condition is typically the case inthe interiors of massive stars.

Massive stars show convection in the core.


ad3B

)()(

π16 3

TrmrL

mGcak

u

I the opacity is large. This is satisfied in the outer oartsof relatively light stars on the MS.

”cold” stars show convection in the outher part.


ad3B

)()(

π16 3

TrmrL

mGcak

u

If this term is large… this is also satisfied in the outer parts of relatively ”cool” stars.

Low mass stars show convection in the outer part.


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16

ad3B

)()(

π16 3

TrmrL

mGcak

u

If the adiabatic gradient is small. This is satisfied in the ionization zone of hydrogen

”cool” stars show convection in the outer part.


sun M2

sun M,51

sun M1

sun M,50

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IonizationHe++, He+, H+

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3D - hydrodynamicalsimulations

3D - hydrodynamicalsimulations(M. Miesch)

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Trampedach (2009)

Star Betelgeuse in Orion Mass 5 M(sun) Radius Ca. 600 R(sun) = 3 AELuminosity 41400 L(sun) Simulation 7.5 yr

Bernd FreytagUppsala Universitet in Sweden

Star Betelgeuse in Orion Mass 5 M(sun) Radius Ca. 600 R(sun) = 3 AELuminosity 41400 L(sun) Simulation 7.5 yr

TemperatureStar Betelgeuse in Orion Mass 5 M(sun) Radius Ca. 600 R(sun) = 3 AELuminosity 41400 L(sun) Simulation 7.5 yr

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20

rrT

rTT

dd

dd

ad

(6.21)

Estimate of superadiabatic temperature gradient T

rrT

rTT

dd

dd

ad

Convective heat flux:

TcvF P con (6.22)


surrounding receivesenergy u per unit volume

mean vertical velocity of convective element

rrT

rTT

dd

dd

ad

TcvF P con

Small: convection isvery efficient


Convective heat flux:

Estimate of convective velocity v

Equate kinetic energy/volume to buoyancy work over distance r

Introduce dimensionless measure for superadiabatic temperature gradient

TcvF P con

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21


With

Convective luminosity Lcon=4R2Fcon

With

and …. total internal energy of star


Simple interpretation with~1

where is measure of the transported internal energy excess

and is the convective timescale tcon

tcon is a dynamical timescale but increased by , because onlydifference in density provides force!


Similar to radiative transport, where a sufficiently large T-grad wasrequired, the dimensionless T-gradient has also to be sufficientlylarge for driving convection:

With and

For solar values of tdyn and tKH and for r/R=0.1 we obtain

Extremely small temperature gradient is sufficient to drive convection

addd

dd

rT

rT

If convection is present (~ 0):

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22

addd

dd

rT

rT

addd

dd

rT

rT

ad3B

)()(

π16 3

TrmrL

mGcak

u

Energy transport by efficient convection

addd

dd

rT

rT

Energy transport by inefficient convection

Needs to be estimated by means of a convection model, e.g. mixing-length model (e.g., Boehm-Vitensen 1956).

A convection model can in principle be interpreted as aninterpolation formulae for Fc between efficient (d~0) and inefficient (d>0) convection (Gough & Weiss 1976).

S … measure of convective efficacy

Fc

convective heat flux

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23

Convective timescale

Convection zones are chemically homogeneous !This has consequences on the evolution of stars.

<< tnuc

Convective momentum flux(turbulent pressure)

Additional to the convective heat flux Fc, convection also transportsthe flux of momentum pt, which has dimensions of pressure:


Additional to the convective heat flux Fc, convection also transportsthe flux of momentum pt, which has dimensions of pressure:

Sun


Sun

Momentum equation of stellar (envelope) structure:

Documents

Energy transportastro.phys.au.dk/jcd/astrofysik2/Lecture04_convection_handouts.pdf06/09/2013 1 Energy transport • Radiation (photons) • Convection (gas cells) • Heat transport