The model Idealized greenhouse model

the model find values of ts , ta allow outgoing radiative power, escaping top of atmosphere, equal absorbed radiative power of sunlight. when applied planet earth, outgoing radiation longwave , sunlight shortwave. these 2 streams of radiation have distinct emission , absorption characteristics. in idealized model, assume atmosphere transparent sunlight. planetary albedo αp fraction of incoming solar flux reflected space (since atmosphere assumed totally transparent solar radiation, not matter whether albedo imagined caused reflection @ surface of planet or @ top of atmosphere or mixture). flux density of incoming solar radiation specified solar constant s0. application planet earth, appropriate values s0=1366 w m , αp=0.30. accounting fact surface area of sphere 4 times area of intercept (its shadow), average incoming radiation s0/4.

for longwave radiation, surface of earth assumed have emissivity of 1 (i.e., earth black body in infrared, realistic). surface emits radiative flux density f according stefan-boltzmann law:

f
=
σ

t

4




{\displaystyle f=\sigma t^{4}}

where σ stefan-boltzmann constant. key understanding greenhouse effect kirchhoff s law of thermal radiation. @ given wavelength absorptivity of atmosphere equal emissivity. radiation surface in different portion of infrared spectrum radiation emitted atmosphere. model assumes average emissivity (absorptivity) identical either of these streams of infrared radiation, interact atmosphere. thus, longwave radiation, 1 symbol ε denotes both emissivity , absorptivity of atmosphere, stream of infrared radiation.

idealized greenhouse model isothermal atmosphere. blue arrows denote shortwave (solar) radiative flux density , red arrow denotes longwave (terrestrial) radiative flux density. radiation streams shown lateral displacement clarity; collocated in model. atmosphere, interacts longwave radiation, indicated layer within dashed lines. specific solution depicted ε=0.78 , αp=0.3, representing planet earth. numbers in parentheses indicate flux densities percent of s0/4.



the equilibrium solution ε=0.82. increase Δε=0.04 corresponds doubling carbon dioxide , associated positive feedback on water vapor.



the equilibrium solution no greenhouse effect: ε=0

the infrared flux density out of top of atmosphere:

f
↑=
ϵ
σ

t

a


4


+
(
1
−
ϵ
)
σ

t

s


4




{\displaystyle f\uparrow =\epsilon \sigma t_{a}^{4}+(1-\epsilon )\sigma t_{s}^{4}}

in last term, ε represents fraction of upward longwave radiation surface absorbed, absorptivity of atmosphere. in first term on right, ε emissivity of atmosphere, adjustment of stefan-boltzmann law account fact atmosphere not optically thick. ε plays role of neatly blending, or averaging, 2 streams of radiation in calculation of outward flux density.

zero net radiation leaving top of atmosphere requires:

−


1
4



s

0


(
1
−

α

p


)
+
ϵ
σ

t

a


4


+
(
1
−
ϵ
)
σ

t

s


4


=
0


{\displaystyle -{\frac {1}{4}}s_{0}(1-\alpha _{p})+\epsilon \sigma t_{a}^{4}+(1-\epsilon )\sigma t_{s}^{4}=0}

zero net radiation entering surface requires:

1
4



s

0


(
1
−

α

p


)
+
ϵ
σ

t

a


4


−
σ

t

s


4


=
0


{\displaystyle {\frac {1}{4}}s_{0}(1-\alpha _{p})+\epsilon \sigma t_{a}^{4}-\sigma t_{s}^{4}=0}

energy equilibrium of atmosphere can either derived 2 above equilibrium conditions, or independently deduced:

2
ϵ
σ

t

a


4


−
ϵ
σ

t

s


4


=
0


{\displaystyle 2\epsilon \sigma t_{a}^{4}-\epsilon \sigma t_{s}^{4}=0}

note important factor of 2, resulting fact atmosphere radiates both upward , downward. ratio of ta ts independent of ε:

t

a


=



t

s



2

1

/

4




=



t

s


1.189




{\displaystyle t_{a}={t_{s} \over 2^{1/4}}={t_{s} \over 1.189}}

thus ta can expressed in terms of ts, , solution obtained ts in terms of model input parameters:

1
4



s

0


(
1
−

α

p


)
=

(
1
−


ϵ
2


)

σ

t

s


4




{\displaystyle {\frac {1}{4}}s_{0}(1-\alpha _{p})=\left(1-{\frac {\epsilon }{2}}\right)\sigma t_{s}^{4}}

or

t

s


=


[




s

0


(
1
−

α

p


)


4
σ





1

1
−


ϵ
2





]


1

/

4




{\displaystyle t_{s}=\left[{\frac {s_{0}(1-\alpha _{p})}{4\sigma }}{\frac {1}{1-{\epsilon \over 2}}}\right]^{1/4}}

the solution can expressed in terms of effective emission temperature te, temperature characterizes outgoing infrared flux density f, if radiator perfect radiator obeying f=σte. easy conceptualize in context of model. te solution ts, case of ε=0, or no atmosphere:

t

e


≡


[




s

0


(
1
−

α

p


)


4
σ



]


1

/

4




{\displaystyle t_{e}\equiv \left[{\frac {s_{0}(1-\alpha _{p})}{4\sigma }}\right]^{1/4}}

with definition of te:

t

s


=

t

e




[


1

1
−


ϵ
2





]


1

/

4




{\displaystyle t_{s}=t_{e}\left[{\frac {1}{1-{\epsilon \over 2}}}\right]^{1/4}}

for perfect greenhouse, no radiation escaping surface, or ε=1:

t

s


=

t

e



2

1

/

4


=
1.189

t

e




t

a


=

t

e




{\displaystyle t_{s}=t_{e}2^{1/4}=1.189t_{e}\qquad t_{a}=t_{e}}

using parameters defined above appropriate earth,

t

e


=
255
 

k

=
−
18
 

c



{\displaystyle t_{e}=255~\mathrm {k} =-18~\mathrm {c} }

for ε=1:

t

s


=
303
 

k

=
30
 

c



{\displaystyle t_{s}=303~\mathrm {k} =30~\mathrm {c} }

for ε=0.78,

t

s


=
288.3
 

k



t

a


=
242.5
 

k



{\displaystyle t_{s}=288.3~\mathrm {k} \qquad t_{a}=242.5~\mathrm {k} }

.

this value of ts happens close published 287.2 k of average global surface temperature based on measurements. ε=0.78 implies 22% of surface radiation escapes directly space, consistent statement of 15% 30% escaping in greenhouse effect.

the radiative forcing doubling carbon dioxide 3.71 w m, in simple parameterization. value endorsed ipcc. equation



f
↑


{\displaystyle f\uparrow }

,

Δ
f
↑=
Δ
ϵ

(
σ

t

a


4


−
σ

t

s


4


)



{\displaystyle \delta f\uparrow =\delta \epsilon \left(\sigma t_{a}^{4}-\sigma t_{s}^{4}\right)}

using values of ts , ta ε=0.78 allows



Δ
f
↑


{\displaystyle \delta f\uparrow }

= -3.71 w m Δε=.019. change of ε 0.78 0.80 consistent radiative forcing doubling of carbon dioxide. ε=0.80,

t

s


=
289.5
 

k



{\displaystyle t_{s}=289.5~\mathrm {k} }

thus model predicts global warming of Δts = 1.2 k doubling of carbon dioxide. typical prediction gcm 3 k surface warming, because gcm allows positive feedback, notably increased water vapor. simple surrogate including feedback process posit additional increase of Δε=.02, total Δε=.04, approximate effect of increase in water vapor associated increase in temperature. idealized model predicts global warming of Δts = 2.4 k doubling of carbon dioxide, consistent ipcc.