meto 621 lesson 19. role of radiation in climate we will focus on the radiative aspects of climate...

METO 621

Lesson 19

Role of radiation in Climate

• We will focus on the radiative aspects of climate and climate change

• We will use a globally averaged one dimensional radiative-convective approximation.

• First we will assume that the atmosphere has negligible absorption for visible radiation

• Then we will add visible absorption

Radiative Equilibrium with Zero Visible Opacity

• The surface is assumed to be reflective in the visible and black in the IR.

• Thus the surface is heated by incoming solar radiation and by downwelling IR radiation from the atmosphere

• The atmosphere is heated by IR radiation from the surface and the surrounding atmospheric layers. This will set up a diffusive-like temperature gradient throughout the optically thick region.

• At the upper ‘edge’, when the optical depth drops to 1 the atmosphere radiates to space, at a globally averaged effective temperature, Te ,determined by the overall energy balance

• Also assume that the optical depth is independent of frequency – the gray approximation.

Zero Visible Opacity

(1) )(),(),(

then,0 , scattering no assume weif and

)()()(

can write wesource thermalaFor

255K. is re temperatueffective the

Earth For the constant.Boltzmann -Stefan theis and

albedo spherical theis constant,solar theis where

4

)1(

4

4/1

ττττ

πτσνττ

σρ

σρ

ν

BuId

udIu

a

TdBBS

S

ST

B

B

Be

−=

=

==≡

⎥⎦

⎤⎢⎣

⎡ −=

∫


[ ]

[ ] ),(),(

),(),(

)(2),(),(),(),(

equations thesegsubtractin and adding

)(),(),(

)(),(),(

are equations range-half The

μτμττ

μτμτμ

τμτμττ

μτμτμ

τμττ

μτμ

τμττ

μτμ

−+−+

−+−+

−−

++

−=+

−+=−

−=−

−=

IId

IId

BIId

IId

BId

dI

BId

dI


• In radiative equilibrium the net flux Fτ is equal to the net outgoing flux , σBTe

4 , which is constant for all τ.

• If we integrate the equation 1 over solid angles then we get

0)(4 =−= BIddF πτ

• In radiative equilibrium, the source function is equal to the mean intensity.

⎥⎦

⎤⎢⎣

⎡+== ∫ ∫∫ −+

−

1

0

1

0

1

1

),(),(2

1),(

2

1)( μτμμτμττ IdIduduIB


[ ] ∫∫

∫ ∫

==−=

⎥⎦

⎤⎢⎣

⎡+−=

−+

−+

1

0

41

0

1

0

1

0

2),(),(2

mequilibriu radiativein that noting

),(),(2

1),(

),(

becomesequation transfer radiativegray theThus

constT

dIIdì∂F

IdIduId

udIu

eB

π

σμμπμτμτμ

μτμμτμττ

τ

2 .................. )(

)(2)(

)( with ),( replace

Weion.approximat stream twoapply the nowcan We

−+−+

−+−+

−=+

−+=−

IId

IId

BIId

IId

BB

τμ

ττ

μ

τμτ


[ ]

C2

)(

give tointegrated becan This

constant 2

)(

get we,T2))(I-)((I 2F

of in terms side handright theexpressing and

2,equation of side handleft theinto )I(I )( inserting

2)()(2F

constraint with thesolved bemust which

4

4

4eB

-

-

4

+=

==

==

+=

=−=

+

+

−+

τμπ

στ

μπ

σ

τ

τ

σμττμπ

τ

σμττμπ

eB

eB

eB

TB

T

d

dB

B

TII


[ ]

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧+−=

−=

+=+=

−==

=

−+

−−

μτσσ

π

πσσ

τττμπ

στ

σμτμπτ

σμττ

*12

2

1

and2

)2(

*)(*)(2

1*

2*)(

medium theof bottom at thefunction source theevaluate weIf

)(2*)(2*)(

get flux we downward for the Solving .2 is surface

at theflux upward the*At n.integratio ofconstant a is C

44

44

4

44

4

eBSB

eBSB

eB

eSB

eB

TTC

TT

IICT

B

TTIF

T


factor greenhouse surface thecalled is where

)2/*1(

get weaboveequation first in the 0 setting2

)0( hence 2)0(2)0(

)0(2

1)0( and ,definitionby 0)0(

0at look uslet Now

)*(12

2

1)(

get we thusand

4/14/1

44

44

G

GTTT

TBTIF

IBI

TTB

eeS

eBeB

eBSB

≡+=

=

===

==

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡ −+−=

+

+−

μττ

πσσμμπ

τμττσσ

πτ

Greenhouse effect - one atmospheric layer model

• Є is the fraction of the IR radiation absorbed by the atmosphere


re. temperatucatmospheri for the

expression mequilibriu radiative theasknown is This

22

1)(

given aat re temperatufor theother The

)1(2

)(

function sourceblackbody for the One

algebra'. simpleafter ' before defined equations

thefrom derived becan sexpressionother Two

4/1

4

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

+=

μττ

τμτ

πστ

ere

eB

TT

TB

Finite Visible Opacity• Any realistic atmosphere absorbs radiation at both IR and

UV/visible wavelengths. • The procedure to solve this problem is similar to that for the

case of no visible opacity.• I will give only the solutions, using the two stream

approximation.

[ ] μμγγγγγ

ττ

μ

πτ

δτδ

πττ

μγτ

μτ

/ where)1()1(2

1)()(

day. aover anglezenith solar average theis and ,/ of ratio

theis cycle, diurnal aover flux solar average theis where

4)(

4

1)()(

0/2

4

4

0

/ 0

neT

TG

kk

nF

en

FI

FIB

e

VISIR

Sa

nS

aIR

≅++−==

+=−=

−

−−−

Finite Visible Opacity

VIR kk >>>> lyequivalent or 1γ

• This is the strong greenhouse limit where the solar radiation penetrates deeply into the atmosphere, In the deep atmosphere, the greenhouse enhancement “saturates” to the constant value G(τ→∞) = (1+γ)/2

• The asymptotic temperature is 4/1

0

4/1

0

22

1)*( ⎟⎟

⎠

⎞⎜⎜⎝

⎛≈⎟⎟

⎠

⎞⎜⎜⎝

⎛+=∞→

V

IRee k

kT

nTT

μμμτ

• This solution resembles that for Venus, which has a surface temperature of 800 K. It does not apply to the Earth or Mars, because of the importance of the surface in the radiative transfer, and the neglect of convection.



• For this case γ1 • This represents an isothermal situation where the

solar heating exactly balances the IR escape.


• For the case γ1 or kIR<<kV

• This represents the anti-greenhouse case. This is relevant to numerous phenomena in the solar system

• An inverted temperature structure characterizes the Earth’s upper atmosphere, where high middle-UV opacity due to ozone absorption gives rise to a temperature inversion.

• This scenario may have happened in the Earth’s history. Worldwide cooling causing mass extinction as the result of an injection of massive quantities of dust (meteoroid impact)

• Stratospheric aerosols (τ up to 10), from Mt Toba eruption some 70,000 years ago may have been responsible for a subsequent cooling of the Earth for a period of 200 years.

meto 621 lesson 19. role of radiation in climate we will focus on the radiative aspects of climate...

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