Radiative Equilibrium

• Equilibrium state of atmosphere and surface in the absence of non-radiative enthalpy fluxes

• Radiative heating drives actual state toward state of radiative equilibrium

1

Figure by MIT OpenCourseWare. 2

Extended Layer Models

4 4

2 2

4 4 4 4 4

1 2

4 4 4

1

1 14 4

1

:

: 2

:

3 2

e e

s e s

s e

s e e

TOA T T T T

Middle Layer T T T T T

Surface T T T

T T T T

3

Effects of emissivity<1

4 4 4

1

14

4 4

2

14

: 2

5 3212

: 2

1 2142

A A A A s

A e s

t t t

t e e

Surface T T T

T T K T

Stratosphere T T

T T K T

4

Full calculation of radiative equilibrium

5

Figure by MIT OpenCourseWare. 6

Time scale of approach to equilibrium

7

8

Contributions of various absorbers

Note: All simulations have variable clouds interacting with radiation

9

10

Problems with radiative equilibrium solution

• Too hot at and near surface

• Too cold at a near tropopause

• Lapse rate of temperature too large in the troposphere

• (But stratosphere temperature close to observed)

11

Missing ingredient: Convection

• As important as radiation in transporting enthalpy in the vertical

• Also controls distribution of water vapor and clouds, the two most important constituents in radiative transfer

12

13

When is a fluid unstable to convection?

• Pressure and hydrostatic equilibrium

• Buoyancy

• Stability

14

Hydrostatic equilibrium

1

:

:

:

,

Weight g x y z

Pressure p x y p p x y

dwF MA x y z g x y z p x y

dt

dw psg

dpecifi

t zc volume

15

Pressure distribution in atmosphere at rest

0

*: ,

1:

: , " "z

H

RT RIdeal gas R

p m

p gHydrostatic

p z RT

RTIsothermal case p p e H scale height

g

Earth: H~ 8 Km 16

Buoyancy

:

:

:

b

b b

be

eb

e

Weight g x y z

Pressure p x y p p x y

dwF MA x y z g x y z p x y

dt

dw p p gg but

dt z z

dwg B

dt

17

Buoyancy and Entropy

Specific Volume: 1

Specific Entropy: s

( , )p s

p

p s

Ts s

s p

p

ss

g T Tg s s

p zB s

18

The adiabatic lapse rate

:

: 0

: 0

revv

v

v

p

p

p

ds p

First Law of Thermodynamics

ds dT dQ T c p

dt dt dt

d pdT dpc

dt dt dt

dT dpc R

dt dt

dT dpc

dt dt

Adiabatic c dT dp

Hydrostatic c dT gdz

gdTdz c

19

p

gc

Earth’s atmosphere: 1

100K

m

20

Model Aircraft Measurements (Renno and Williams, 1995)

This image (published on Monthly Weather Review by the American Meteorological Society) is copyright © AMS and used with permission. 21

Radiative equilibrium is unstable in the troposphere

Re-calculate equilibrium assuming that

tropospheric stability is rendered neutral by convection:

Radiative-Convective Equilibrium

22

Better, but still too hot at surface, too cold at tropopause 23

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