Lesson 2

AOSC 621

Radiative equilibrium

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The total amount of solar radiation intercepted by the Earth is πRe2,

where Re is the radius of the earth. This energy will be spread over

the entire area of the Earth - 4πRe2 . Hence the average solar flux

that reaches the top of the atmosphere (TOA) is given by :

FS = πRe2S /4πRe

2 = S /4

Where S is the solar constant. The earth reflects some of this radiation . Let the albedo be ρ, then the energy absorbed by the earth is (1- ρ)Fs.

The thermal energy emitted by the earth is σBTe4

Hence we get 4/1

)1(⎥⎦⎤

⎢⎣⎡ −

=B

se

FTsρ

Radiative equilibrium• The spherical albedo of the earth is about 30%. This

gives an effective temperature of 255 K. This is considerably lower than the measured average temperature at the surface Ts of about 288 K.

• The fact that Ts>Te is seen in all of the terrestrial planets with substantial atmospheres. It is due to the fact that whereas the planets are relatively transparent in the visible part of the spectrum, they are relatively opaque in the infra-red.

• A portion of the IR emitted by the surface and atmosphere is absorbed by polyatomic gases and re-emitted in all directions – including back to the earth.

Greenhouse effect - one atmospheric layer model

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

Greenhouse effect

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Consider the radiative equlibrium in the atmosphere.

The amount of energy absorbed is εσ bTE4 and the amount

emitted is 2σ bTA4 . Hence in equilibrium:

TA4 = TE

4 /2Now consider the energy balance at the surface :

(1− ρ )S4

+ εσ bTA4 =σ bTE

4

substituting for TA we get

(1− ρ )S4

+ εσ bTE4

2=σ bTE

4

which gives

TE = (1− ρ )S4σ b (1−ε /2)

⎡ ⎣ ⎢

⎤ ⎦ ⎥

1/ 4

Greenhouse effect

• An examination of the previous equation shows that as e increases the ground temperature increases.

• This is what is referred to as greenhouse warming.• As we shall show later, not all of the additional

energy trapped by the additional absorption goes inot heating the surface. Some , for example goes into evaporation of water - which stores energy in the atmosphere as additional latent heat.

• So one should view the inference of the equation with some caution.

• This issue will be discussed later when climate feedback is addressed.

Radiative equilibrium• First Law of Thermodynamics - the time rate of change

of the column energy is

TOAs FFNtE

−−==∂∂

)1( ρ

• and4SBeffTOA TF sT=

• Where N is called the radiative forcing

Where Teff is the effective transmission of the atmosphere to thermal radiation ( (1-e) in the simple model)

Perturbation to the radiative forcing

• What happens if N(TS) is changed by Δ N(TS) • Assume that the atmosphere moves to a new

equilibrium temperature, then we can write

0=Δ∂∂

+Δ ss

TTNN

We define the climate sensitivity α as

NT ds Δ=Δ α

Climate sensitivity

⎥⎦⎤

⎢⎣⎡

∂−∂

−∂∂

=∂∂−= −

s

s

s

TOAs T

FTFTN

v)1()/( 1 ρα

CO2 doubling

• The calculated radiative forcing from doubling CO2 is ~4 Wm-2

• Assuming that the solar flux is constant we get

{ }TOA

seffsB

s

effsB

FTTT

TTT

44

)( 13

14

==⎪⎭⎪⎬⎫

⎪⎩⎪⎨⎧

∂∂

=−

−

σσ

α

• FTOA= 240 Wm-2 and Ts = 288 K, hence α=0.3 Wm-2K-1

• Hence the increase in surface temperature is 1.2 K

Change in the solar constant• Suppose the solar constant were to decrease by 1%• Assume no feedbacks other than the reduced thermal

emission. If there is no albedo change due to the decreased solar flux then the second term in the climate sensitivity is zero. Hence we get:

{ }

SS

S

sTOA

FTFNTF

Δ−=ΔΔ−=Δ∂∂=

)1()1(

/

ραρ

α

Change in the solar constant 2

• Substituting for FTOA

S

Ssds

TOAss

sBeff

FFTT

FTT

TT

Δ⎭⎬⎫

⎩⎨⎧=Δ

⎭⎬⎫

⎩⎨⎧

=⎪⎭⎪⎬⎫

⎪⎩⎪⎨⎧

∂∂

=−−

4

4(114s

α

• Change in temperature is -0.72 K

General expression for α• Consider a parameter Q ( such as albedo) that depends

on the surface temperature• The direct forcing ΔN is augmented by a term

€

∂N∂Q

. ∂Q∂Ts

.ΔTs

• Solving for the climate sensitivity we find

1)}/)(/(/4{ −∂∂∂∂−= ssTOA TQQNTFα

General expression for α

• If we have an arbitrary number of forcings Q then one can derive a general expression for the climate sensitivity

[ ] 1)//()/(/4 −∑ ∂∂∂∂−= ssTAO TQQNTFα

It should be noted that the above equation assumes that the individual forcings are independent of one another

Radiative forcing of greenhouse gases

Radiative forcing

Negative feedback

Positive feedback

Polar ice coverage - positive feedback

CLIMATE FEEDBACK MECHANISMS

• ·POSITIVE AND NEGATIVE FEEDBACKS

• ·WATER VAPOR - POSITIVE

• ·ICE COVER - POSITIVE

• .CLOUDS - POSITIVE AND NEGATIVE - MAINLY NEGATIVE

Fig. 7-27, p. 191

Positions of the jet streams

Fig. 8-30, p. 231

Relation between jet stream and high and low pressure systems

Using Total Ozone measurements• Contributions to the column ozone amount

(total ozone) come mainly from the lower stratosphere. In this altitude region ozone has a long chemical lifetime compared to its dynamic lifetime. Hence one can treat Total Ozone acts as a dynamic tracer.

• The subtropical and polar fronts correspond to a rapid change in the tropopause height, and total ozone shows abrupt changes.

• Hudson et. al, 2003, 2006 developed a method to determine the position of the fronts (jet streams) from satellite total ozone data.

Jet Streams on March 11, 1990Sub-tropical = Blue, Polar = Red

Mean latitude of the Sub-tropical front

Shift from 1980 to 2004 is 1.7 degrees

Mean Latitude of the Polar Front

Shift from 1980 to 2004 is 3.1 degrees

Summary of values of the tropical belt expansion

• Zonal mean stream function study gives values from 2.5 to 3.1 degrees

• Tropopause analysis gives values of 5-8 degrees• OLR study gives values from 2.3 to 4.5• Ozone gives value for the Northern Hemisphere only of

1.7 degrees• All over the period 1979 to 2005

21st Century Predicted WideningEnsemble mean ~2 deg. latitude Lu et al. (2007)

south Hadley Cell north

Northern Edge

Southern Edge

Observed vs Predicted Widening

• Observed: ~2-5°latitude 1979-2005• Modeled (A2 scenario): ~2°lat over 21st century• Modeled 25-yr trends during 21st century

– 40 100-yr projections → – Mean 0.5°– >2.0° in only 3% of cases– Max 2.75°– (Celeste Johanson, Univ. of Washington)

Temperature deviation (GISS) versus latitude

Number of tropical pixels versus time for longitude -80 to -60

28 to 32 latitude

48 to 52 latitude

38 to 42 latitude

58 to 62 latitude

Number of tropical pixels for 58 to 62 latitude vs longitude