From Intensity to Flux
¯
Fsurf
=
2p
1
òò ò
¯
I surf
(m, f, l )m dm df d l
l f =0 m =0
• ϕ integral: Azimuthal Symmetry in longwave. (A little more
complicated in shortwave)
• μ integral: We’ll replace intensities with monochomatic flux and
work at a single zenith angle . This is often called the “Diffusivity
Approximation”
• λ integral: This is one is a pain. We can either do it explicitly (lineby-line) or use various approximations. We’ll discuss this later.
• Really 4 integrals, as the intensity term contains an integral over
the emission from each atmospheric level z !!
Diffusivity Approximation (μ integral)
(aka 2-stream Approximation)
• We define an effective angle
approximately correct:
such that the “flux transmittance” is
• For ANY τ(z1,z2), can find a value of
need to find a general one.
such that the equation is exact. But we
• Typical values are 1/2 to 2/3. Typical flux errors are 1-3%.
• With this approximation plus using azimuthal symmetry, we get flux equations
that look exactly like intensity equations:
F (l ) =
¯
surf
where
TOA
ò
z=0
p B(z)WF¯(z)dz
Key Assumptions for pure radiative equilibrium
• Each layer in atmosphere in pure radiation balance: Flux in = Flux Out for
each thin layer. Same for surface.
• Diffusivity/2-stream approximation (including azimuthal average): Can
ignore angular integral and use one specific direction (typically theta = 48
deg).
• For us, we’ll further assume:
• Atmosphere does not absorb solar
• All solar absorbed by surface only
• In the longwave, “Graybody approximation” Gas absorption is
independent of wavelength. (Gets around the wavelength integral!)
• Manabe & Strickland (1964) relaxed the latter set of assumptions and
calculated pure Rad. Eq. temperature profiles for real gas behavior,
including some Shortwave absorption in the atmosphere.
• The basic behavior will be qualitatively correct.
Revisiting the Radiative Transfer Equation
OLR=Fup(TOA)=Fsol
Fsol = So (1-)/4
T1
T2
Ts4
*
T3
Ts
Basic Question: what temperature profile is required to sustain
an energy balance (i.e. a radiative equilibrium) at the TOA,
surface and for each layer of atmosphere)?
Methodology
• Write Radiation balance for every layer and surface.
• Apply BC’s at TOA and surface.
• Make assumption about absorption coefficient vs. altitude.
Solution to the Gray-Gas Model
Discontinuity at
surface, FSOL/2
FSO
0
L
F
F
*
T4
FSOL
2
A “tropopause” (skin) temperature for our simple model
1
Tstrat
1
æ FSOL ö 4 æ 239 2 ö 4
=ç
÷ »ç
÷ = 214K
è 2s ø
è 5.68´10-8 ø
Solutions to the Gray Gas Model
This reduces to:
Hypothesize that:
(z)h2o(z) (which decays
exponentially in height)
FSOL æ 3 * -z H ö
s T ( z) =
ç t e +1÷
ø
2 è2
4
(z)  *exp(-z/H)
Scale height of H2O, ~2.3 km
Gray body model for the
atmospheric temperature
profile:
6.5 K/km lapse rate
Temperature profiles from pure radiative equilibrium:
http://ocw.mit.edu/courses/earth-atmospheric-and-planetary-sciences/12-811-tropical-meteorology-spring2011/lecture-notes/