meto 621 lesson 19. role of radiation in climate we will focus on the radiative aspects of climate...
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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
−=
=
==≡
⎥⎦
⎤⎢⎣
⎡ −=
∫
Zero Visible Opacity
[ ]
[ ] ),(),(
),(),(
)(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
Zero Visible Opacity
• 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
Zero Visible Opacity
[ ] ∫∫
∫ ∫
==−=
⎥⎦
⎤⎢⎣
⎡+−=
−+
−+
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
Zero Visible Opacity
[ ]
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
Zero Visible Opacity
[ ]
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧+−=
−=
+=+=
−==
=
−+
−−
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
Zero Visible Opacity
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
Zero Visible Opacity
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
Zero Visible Opacity
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.
Finite Visible Opacity
Finite Visible Opacity
• For this case γ1 • This represents an isothermal situation where the
solar heating exactly balances the IR escape.
Finite Visible Opacity
• 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.