Radiative heat transfer

Radiative heat transfer

Radiative heat transfer

Black body radiation

Radiative heat transfer

Black body radiation

Spectral intensity (W.m-2 µm-1 )

P_e( , T) = 2⇡hc²

5

1

exp(hc/ k_BT) 1

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Power emitted per unit surface :

Radiative heat transfer

Black body radiation

Spectral intensity (W.m-2 µm-1 )

P_e( , T) = 2⇡hc²

5

1

exp(hc/ k_BT) 1

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Power emitted per unit surface :

M(m) = 2.88 ⇥ 10 ³ T

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Wien’s displacement law

Radiative heat transfer

Black body radiation

P_BB(T) =

Z ₁

0

P_e( , T)d = 2⇡⁵k_B⁴

15 h³c²T⁴ = T⁴

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Stefan’s law

Spectral intensity (W.m-2 µm-1 )

P_e( , T) = 2⇡hc²

5

1

exp(hc/ k_BT) 1

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Power emitted per unit surface :

M(m) = 2.88 ⇥ 10 ³ T

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Wien’s displacement law

Radiative heat transfer

Black body radiation

P_BB(T) =

Z ₁

0

P_e( , T)d = 2⇡⁵k_B⁴

15 h³c²T⁴ = T⁴

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Stefan’s law

Spectral intensity (W.m-2 µm-1 )

P_e( , T) = 2⇡hc²

5

1

exp(hc/ k_BT) 1

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Power emitted per unit surface :

Non black bodies

Emissivity ε = absorptivity a P(λ,T) = P_e(λ,T) ε(λ)

Kirchoff’s law

M(m) = 2.88 ⇥ 10 ³ T

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Wien’s displacement law

Radiative heat transfer

Black body radiation

Grey bodies

Emissivity independent of λ ε(λ) = a(λ) = C

P_BB(T) =

Z ₁

0

P_e( , T)d = 2⇡⁵k_B⁴

15 h³c²T⁴ = T⁴

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Stefan’s law

Spectral intensity (W.m-2 µm-1 )

P_e( , T) = 2⇡hc²

5

1

exp(hc/ k_BT) 1

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Power emitted per unit surface :

Non black bodies

Emissivity ε = absorptivity a P(λ,T) = P_e(λ,T) ε(λ)

Kirchoff’s law

M(m) = 2.88 ⇥ 10 ³ T

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Wien’s displacement law

Diameter of the Sun: 1,4 10⁶ km

Surface temperature of the Sun : 5800 K Distance to the Sun: 1,5 10⁸ km

Diameter of the Earth: 12700 km

Effective albedo of the Earth: 0.31

The Donald Trump problem at zeroth order :

show that with a completely transparent atmosphere, it would be difficult to grow lawn for a golf course on Earth.

Earth and Sun are assumed to be black bodies

-19°C

Diameter of the Sun: 1,4 10⁶ km

Surface temperature of the Sun : 5800 K Distance to the Sun: 1,5 10⁸ km

Diameter of the Earth: 12700 km

Effective albedo of the Earth: 0.31

The Donald Trump problem at zeroth order :

show that with a completely transparent atmosphere, it would be difficult to grow lawn for a golf course on Earth.

Earth and Sun are assumed to be black bodies Radiative flux from the sun, averaged over entire Earth surface : 342 W/m²

Flux adsorbed : 235 W/m²

Corresponding equilibrium temperature : 254°K

The Donald Trump problem at order one :

show that with a one layer atmosphere adsorbing in the infrared, it would be easier to grow lawn for a golf course on Earth.

Incoming solar radiation Reflected solar radiation

Atmosphere emissivity ε

Ta

Te Earth surface emissivity 1

Write the energy flux balance for the earth + atmosphere system Write the energy flux balance for the atmosphere at temperature Ta

Determine Ta and Te as a function of ε

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

emissivity

210 220 230 240 250 260 270 280 290 300 310

Temperature

Tearth Tatmos

T

=

 (1 a)J

(1 ✏/2)

1/4

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

The Donald Trump problem at order 2 :

Evaluate the temperature distribution in a stable « grey » non scattering atmosphere

Atmosphere clear to solar radiation ; all absorption takes place at the ground.

Gray atmosphere with no clouds in the IR (no scattering of light) Layers of atmosphere are grey thermal radiators

Plane parallel or stratified atmosphere

Eddington approximation for the IR emission : I(z,θ ) = I0(z) + I1(z) cos θ

z

z+dz θ

s

ds

I(s) I(s+ds)

s

ds

I(s) I(s+ds)

Extinction (absorption, scattering)

dI

= 

I ds

s

ds

I(s) I(s+ds)

Extinction (absorption, scattering)

dI

= 

I ds

Augmentation (emission, scattering)

dI

= 

J ds

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

s

ds

I(s) I(s+ds)

If no scattering, emission by thermal radiation

dI

= 

T

⇡ = 

B (T )

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Extinction (absorption, scattering)

dI

= 

I ds

Augmentation (emission, scattering)

dI

= 

J ds

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

s

ds

I(s) I(s+ds)

If no scattering, emission by thermal radiation

dI

= 

T

⇡ = 

B (T )

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Extinction (absorption, scattering)

dI

= 

I ds

Augmentation (emission, scattering)

dI

= 

J ds

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

dI

ds = 

( I + B )

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

dI

ds = 

( I + B )

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

dI

ds = 

( I + B )

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

d⌧

= 

ds

Optical path length (dimensionless) τ

dI

ds = 

( I + B )

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

d⌧

= 

ds

Optical path length (dimensionless) τ

dI

d⌧ = I B

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

J

(1-a)

J

(1-a)

F_down(z)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

J

(1-a)

F_down(z)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

F_up(z)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

J

(1-a)

F_down(z)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

F_up(z)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

⌧ =

Z

z



dz

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Optical path from atmosphere top

Atmosphere top τ = 0

J

(1-a)

F_down(z)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

F_up(z)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

⌧ =

Z

z



dz

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Optical path from atmosphere top

Atmosphere top τ = 0

J

(1-a)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

F_up(z)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

⌧ =

Z

z



dz

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Optical path from atmosphere top

Ground τ = τ*

Atmosphere top τ = 0

J

(1-a)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

F_up(z)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

⌧ =

Z

z



dz

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Optical path from atmosphere top

Ground τ = τ*

Atmosphere top τ = 0

J

(1-a)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

F_up(z)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

⌧

=

Z

0



dz

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Optical depth of atmosphere

⌧ =

Z

z



dz

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Optical path from atmosphere top

Ground τ = τ*

Atmosphere top τ = 0

J

(1-a)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

F_up(z)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

F

= F

(⌧

) = ⇡B (T

)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Radiation going up :

⌧

=

Z

0



dz

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Optical depth of atmosphere

⌧ =

Z

z



dz

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Optical path from atmosphere top

Ground τ = τ*

Atmosphere top τ = 0

J

(1-a)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

F_up(z)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

F

= F

(⌧

) = ⇡B (T

)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Radiation going up :

J

a + F

(⌧

)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Radiation going down :

⌧

=

Z

0



dz

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Optical depth of atmosphere

⌧ =

Z

z



dz

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Optical path from atmosphere top



= 

exp( z/H

)

H

= 1.6 km (water vapor)



= 

exp( z/H

)

H

= 1.6 km (water vapor)

Ground Temp.

T

= J

(1 a)

✓

1 + 3⌧

4

◆



= 

exp( z/H

)

H

= 1.6 km (water vapor)

Ground Temp.

T

= J

(1 a)

✓

1 + 3⌧

4

◆

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

⌧

=

Z

0



dz

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Optical depth of atmosphere



= 

exp( z/H

)

H

= 1.6 km (water vapor)

Ground Temp.

T

= J

(1 a)

✓

1 + 3⌧

4

◆

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

⌧

=

Z

0



dz

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Optical depth of atmosphere



= 

exp( z/H

)

H

= 1.6 km (water vapor)

Steep temperature gradient leads to convective instability

390 W/m²