7. Earth’s climate and radiationZero-dimensional planetary energy budget equation




Equivalent blackbody temperature: TeAlchemy (albedo,Radius of Earth: aRubedo, Nigredo)
Distance between Earth and Sun in astronomical unit: Rps
Pseudo planetary emissivity for surface emission: Planetary albedo: S224a(1)=4a
TSolar constant: Se2
Te  256K, Ts  280K
  0.69

 Te4 =  Ts4
S= 1370 W/m2
 = 0.3

 Qs
 Ts4Rps

Planets and their climate
r= 6.37 x103 km
M=1.99x1030 kg
r= 6.96 x105 km
E= 3.85x1026 W
Rpe=1.50x108 km
S= 1370 W/m2

RpsVenus 0.72
Earth 1.00
Mars
1.52
Jupiter 5.20

Te
0.77 227
0.30 2560.15 216
0.58 98 a 2 (1   )

Te,m
230
250
220
130

Ts (1-g)vs IR
750 3.97 224
280 0.49 1.15
240 0.20 0.83
134 1.59-
0.01
0.64
0.71
0.89

S
2
4=4

a
Te
2
Rps

Radiative energy budget change a 2 (1  Ap )
T
S=4 sSTs

Ts = F

S= 4 a 2  Ts42Rps???

if Ap and  are constant
Cess et al. (1990)

STs
F =, =44(S / 4)

F: Radiative forcing ( ): Equilibrium Climate Sensitivity()Ts=280K, S=1366 W/m2 (Q=S/4= 341.5):
= 0.20 K/(W/m2) cf. GCMs:  = 1
S= 2 W/m2: Ts= 0.1K (GCMs: 0.5K)

 e    

International Satellite Cloud Climatology Project (ISCCP)
 Rossow et al. (1993)
 4 geostational satellites and 2 polar orbitersn( c ,Tc ) Cloud reflectance and clear sky reflectance

tc =1
1 g, rc = 1  t c , b(=   ) =1 + b c2

rp = nrc + (1  n)rg
n = 0.6, rg = 0.1, rp = 0.3
 rc =rp  (1  n)rg

= 0.43ng = 0.85, b = 0.075,  = 1.73

Region
ISCCP SOBS METEOR Nimbus-7
Global
62.6
61.5
60.9
52.9
NH
59.7
59.0
55.7
51.7
SH
65.4
64.0
66.0
54.1
Polar
52.3
68.6
50.4
58.0
Midlatitude 72.2
67.3
68.5
56.9
58.4
55.4
58.2
48.5
Tropics
Land
47.1
53.3
46.5
45.5
Ocean
70.2
65.5
67.9
56.5

11( 1) = 5.8
b 1  rc2r W = e c = 38 g / m 2 (re = 10 μ m)3c =

Two layer emissivity approximation
No absorption of the shortwave radiation in the atmosphere & nodynamical effect Radiative equilibruimF  (0) =  T 4 + (1   ) Tg4 , F  ( 1 ) =  Tg4 , F + ( 1 ) =  T 4
 =  T 4 + (1   ) Tg4 =  Tg4   T 4
21,  T 4 =221 T 4 =  Tg4 , F  (0) =  Te4 = (1  ) Tg422 Tg4 =T = 235K
Te = 256K

conv(F)=0

T < Te < Tg

div(F)=0Tg = 280K

 = 0.6

Heating the surface only
Solar Radiation

TsTerrestrial Radiation

Radiative equilibrium; Radiative-convective equilibrium

(Manabe and Strickler, 1964)

Thermal stratificationof the atmosphere

Cooling and warming byabsorbing gases

Manabe and Wetherald (1967)

 Cloud liquid water path and cloud top temperature

are important parameters to control the earth’s
energy budget.

[n(1  rc ) + (1  n)(1  rs )]Q = nTc4 + (1 n)Tg4

Radiative forcing F= FS + FLFS+FS

FS = FLFL+FLn
T
t=0

F'S = F'Ln+n
Tn'
T't =t

Cloud radiative forcing: CRF = F - F clear= n F cloud+ (1-n) F clear - F clear = n (F cloud+ F clear)
Aerosol direct

IPCC standard

Aerosol indirect effect(IPCC, 1990)

Radiative forcings since 1750
BOA
Cloud and aerosol stillTOA (Tropopause)
LGHG
+0.43

uncertain (±0.5 W/m2)
Large aerosol forcing
at surface

LGHG
+2.63
+0.02 Stratospheric O3 & H2O(-0.13, TAR)

+0.16
+0.21

MIROC

+0.35 Tropospheric O3-1.6-0.5 Aerosol direct-0.5-0.7 Albedo effect(0 to -2 only range, TAR)
-0.1 Surface albedo(-0.2, TAR)

TOA: Values from IPCC/AR4
BOA: Values from MIROC(Takemura et al., 2005)+0.12 Sun(+0.3, TAR)
-0.5 to 1
Cloud forcing dueto 2xCO2(-1 to 0.5, TAR)

Report of Science Council of Japan on global warming problem (2008)

Aerosol direct forcing by MIROC/AGCM+SPRINTARS Uncertainty of about 0.5 W/m2 due to stratification
 Boundary layer modeling and hygroscopic growth modeling

Clear-skyAVG. –0.70 W m-2All-skyAVG. –0.04 W m-2

TOA Radiative
Forcing

Takemura et al. (JGR2005)

References








Hansen, J., M. Sato, R. Ruedy, L. Nazarenko, A. Lacis, G.A. Schmidt, G. Russell, I. Aleinov,M. Bauer, S. Bauer, N. Bell, B. Cairns, V. Canuto, M. Chandler, Y. Cheng, A. Del Genio, G.
Faluvegi, E. Fleming, A. Friend, T. Hall, C. Jackman, M. Kelley, N. Kiang, D. Koch, J.
Lean, J. Lerner, K. Lo, S. Menon, R. Miller, P. Minnis, T. Novakov, V. Oinas, J. Perlwitz, J.
Perlwitz, D. Rind, A. Romanou, D. Shindell, P. Stone, S. Sun, N. Tausnev, D. Thresher, B.
Wielicki, T. Wong, M. Yao, and S. Zhang, 2005: Efficacy of climate forcings. J. Geophys.
Res., 110, D18104, doi:10.1029/2005JD005776.
IPCC, 1990: Climate Change, The IPCC Scientific Assessment. Eds. J. T. Houghton, G. J.
Jenkins, and J. J. Ephraums, Cambridge Univeristy Press, 364 pp.IPCC, 2001: J. T. Houghton, Ed., Cambridge Univ. Press, " Climate Change 2001-The
Scientific Basis".
Manabe, S. and R. F. Strickler, 1964: Thermal equilibrium of the atmosphere with aconvective adjustment. J. Atmos. Sci., 21, 57-81.
Manabe, S, and R.T. Wetheraldm, 1967: Thermal equilibrium of the atmosphere with a given
distribution of relative humidity. J. Atmos. Sci., 24, 241-59. Rossow, W. B., A. W. Walker, and L. C. Garder, 1993: Comparison of ISCCP and othercloud amounts. J. Climate, 6, 2394-2418.

Climate sensitivity 1 Radiation budget imbalance due to a change ( Radiative forcing): F (W/m2)
 Radiative forcing causes a surface airtemperature chage after a time t: Ts (K)

Tg =  (t)F Equilibrium climate sensitivity:  (W/m2K)
 Current GCMs  ~ 1  Thermal inertia efficiency:   = 1 for perpetual forcing  ~ 0.4 for CO2 increase  ~ 0.1 for volcanic erruption aerosolsClimate sensitivity 2
 Perturbation of the systemF =F
FF dqi
e + {+}TS  0eTSi qi dTS

 Radiative forcingFRF =ee Equiliburium climate sensitivityTS =   RF

 Feedback gain factors ( F) (F) dqi
1/  =+= 0 +  iTSqi dTSii

Radiation dampingRadiative energy budget of the planetF = (1   )Qs   Ts4
Total gain factor is the sum of three effects: radiativedampling, planetary albedo change, and emissivity change: = 4Ts3 + Qs
The radiation damping factor causes a strong stablizationof the system

0 =
d
d
+ Ts4= 0 +  + dTsdTs

1
 = 0.64, Ts = 280K
3 = 0.31,
4Ts

If there is no atmosphere, the system is very stable 0,blck = 0.20,  = 1, Ts = 279K

Feedback of water vaporSuppose no-radiation energy convergence atmosphere similar
to the cloud case in visible region.

11+  g / 2

 wv = exp(8.3  2563 / Ts ) + 0.2

Ts4F = (1   )Qs 
1 + 0.83 wv

F = 0  Ts = 288K,  wv = 0.75=

Water vapor can cause a strong instability to the atmosphere

(F)
4Ts3
0 =
=
= 3.3
Ts
1+ 0.83 wv

=

(F) d wv
2127( wv  0.2)Ts 2
 wv ==
= 2.12 wv dTs
(1 + 0.83 wv )1= 0.81 0 +  wv

=0.1 has no solution: Runaway greenhouse effectClimate sensitivity of the GCM Total sensitivity is relatively similar among models.

But, partitioning among the various climate change
factors is very uncertain, for example, between GFDLand GISS GCMs.

IPCC90
Feedbacks
Cloud, albedo fixedCloud feedback
Albedo feedback


GFDL
0.43
0.50
1.00


GISS
0.50
0.80
1.05*)-.).-!,,.%*)*"#,0%.2 (-! +!! *"'%#$.
!(-!
*'.3()*)-.). ! ')&*)-.).  ! -!
! (*'
0*# ,*)/(!,*'/(!*"% !'#-.)  .*(! ( (*'-*'/.!.!(+!,./,!  -*)-.). !#(*'!.!+$)*'.3()*)-.).!(  
*'!/',1!%#$.*" ,2%, #(*'Latent heat of vaporization at 273K
2.500E6 J/Kg
1 bar 10^6 dyne/cm2 = 10^5 N/m2 = 10^5 Pa
Earth's radius
6370 km
Mean solar angular diameter
31.99 minutes of arc
Air= 0.78083 (N2)+0.20947 (O2)+ 0.00934 (Ar)+0.00033 (CO2) by volume ratioGlobe= 0.708 (Ocean)+ 0.292 (Land) by area ratio
Molecular weight of air29 g/mole

Radiative energy balance 1 Equation for broad band thermal radiationconv(F)=0

Heating the surface only
Solar Radiation

div(F)=0

Ts
Terrestrial Radiation

dF +dF +4+
= F + T , =  F  + T 4dxdxd(x)4= (x) + 2T ,dxd(x)= (x)dx

Radiative energy balance 2
div(F) = 0   = C   = 2T , (x) = A  x4

1+F (x) = [A + (1 x)]2
1
F (x) = [A  (1+ x)]2
 Boudary condition

F + (0) = 0,F  (xg ) = Tg4

A = 2Tg4 = (2 + x g )Houghton (1972)

Radiative energy balance 3
 = 2 T 4 = (1 + x) =2 Tg4
 =2 + xg

T4 =

1+ x
2Tg4
2 + xgF + (x) =1+ x 4T
2 + xg g

F (x) =x
Tg4
2 + xg2+ xTg4
2 + xg

 Equivalent black body temperature of the planet

2F (0) =  T = Tg4
2 + xg4e

 There is a temperature discontinuity at surface. Greenhouse effectQ10.
 Calculate the optical thickness of the atmosphere in theinfrared spectrum region with Ts and Te given in Table
1. Radiative energy balance 4More realistic value of the optical thickness of the earth’s
atmosphere From the planetary albedo condition,

A = 0.3  Ag,s = 0.1,n = 0.6,  c ,Vis = 5 c, IR  0.5 c,Vist sky,normal  0.2 or  gas,IR =  ln(tsky, normal) = 1.6

 g  n( c ,IR +  gas,IR / 2) + (1 n) gas,IR = 2.6
Ts = (
2 + xg 1/ 4) Te = 342K, Te = 255K
2This value is similar to Manabe et al. for radiative-equilibrium
model. Upward energy transfer by convection is important to cool
the surface.

      

Dust vail index and paiting colorLamb (1970): (1) the depletion of radiation, (2) temperature variations following
the eruption, and (3) the amount of solid material dispersed as dust after aneruption.
Zerefos et al. (2007): R/G color ratio