Equilibrium Temperature with

Greenhouse Gases

z

T

Equation for temperature change with altitude in the troposphere

Γ is called the lapse rate for the troposphere.

km

Co

7.6

Average lapse rate for dry air in the troposphere

At this altitude, 95% of IR escapes to space without further absorption.

We want to find what the altitude is for 95% escape of IR,

And the surface temperature of the Earth with greenhouse gases.

We will use a two system model of the atmosphere. The two systems being the troposphere and the stratosphere.

We will assume thermal equilibrium in the stratosphere.

Stratosphere

Troposphere

4

tabsTI

4

tabsTI

4)1(

tradTI

4

tabsTI

4)1(

tradTI

4

sradTI

4

sradTI

For thermal equilibrium, power in = power out

Pin=Pout

For Pin we need to multiply all the intensities going in by area

For Pout, we need to multiply all the intensities going out by area.

4

tabsTI

4)1(

tradTI

4

sradTI

4

sradTI

Pin = A(ασTt4+(1- α)σTt

4)

4

tabsTI

4)1(

tradTI

4

sradTI

4

sradTI

Pin = A(ασTt4+σTt

4- ασTt4)

4

tabsTI

4)1(

tradTI

4

sradTI

4

sradTI

Pin = A(ασTt4+σTt

4- ασTt4) = AσTt

4

4

tabsTI

4)1(

tradTI

4

sradTI

4

sradTI

Pout = A(2εσTs4+(1- α)σTt

4)

Pin = Pout

AσTt4 = A(2εσTs

4+(1- α)σTt4)

ε=α, SO

AσTt4 = A(2ασTs

4+(1- α)σTt4)

σTt4 = 2ασTs

4+ σTt4 - ασTt

4

σTt4 = 2ασTs

4+ σTt4 - ασTt

4

0= 2ασTs4- ασTt

4

0= 2Ts4- Tt

4

0= 2Ts4- Tt

4

+Tt4 +Tt

4

Tt4 = 2Ts

4

4

1

2

t

s

TT

The temperature at which IR is escaping has to be 255 K because we have thermal equilibrium at this altitude with no absorption by greenhouse gases.

Thus Tt = 255 K

4

1

2

255 KT

s

And the answer is

Ts = 214 K

214K

214K

z

T

Now let us find the altitude at which Tt = 255 K

km

C

zz

TT o

if

if7.6

)(7.6if

o

ifzz

km

CTT

The average top of the troposphere is zf = 11 km, so

)11(7.6214i

o

izkm

km

CTK

i

o

zkm

CKKK 7.67.73255214

i

o

zkm

CKK 7.67.7341

+73.7K +73.7K

iz

km

KK 7.67.32

km

K

Kz

i

7.6

7.32

And the answer is

zi = 4.9 km

HOMEWORK:

1. Use Tf = 214 K and zf = 11 km to find the surface temperature of the Earth with greenhouse gases.

If CO2 doubles in the atmosphere, the top of the troposphere is predicted to rise to 11.6 km. What would the surface temperature of the Earth be then?