EART164: PLANETARY
ATMOSPHERES
Francis Nimmo
F.Nimmo EART164 Spring 11
Last Week – Clouds & Dust
•
•
•
•
•
Saturation vapour pressure, Clausius-Clapeyron
Moist vs. dry adiabat
Cloud albedo effects – do they warm or cool?
Giant planet cloud stacks
Dust sinking timescale and thermal effects
dP s
dT
t

L H Ps
RT
2
H
gr  
2
F.Nimmo EART164 Spring 11
This Week – Radiative Transfer
• A massive (and complex) subject
• Important
– Radiative transfer dominates upper atmospheres
– We use emission/absorption to probe atmospheres
• We can only scratch the surface:
–
–
–
–
–
Black body radiation
Absorption/opacity
Greenhouse effect
Radiative temperature gradients
Radiative time constants
F.Nimmo EART164 Spring 11
Sounding planet atmospheres
• Penetration depends on wavelength (other things
being equal). Everyday example?
• Absorption is efficient when particle size exceeds
wavelength
• Example at giant planets
Increasing wavelength
Increasing depth
UV
vis/NIR
Thermal IR
radio
~nbar
H2 absorption
~1 bar
Clouds, aerosols
~few bar (variable)
CH4 etc
~10 bar
NH3
F.Nimmo EART164 Spring 11
Absorption vs. Emission
I
I
l
l
cold
warm
warm
cold
• Absorption vs. emission tells
us about vertical temperature
structure
• This is useful e.g. for
exoplanets
Jupiter spectrum
Lissauer & DePater Section 4.3
F.Nimmo EART164 Spring 11
Definitions
Il
Fl 
+
Il
• Intensity (Il) is the rate at which energy
having a wavelength between l and l+l
passes through a solid angle (W/m m-2 sr-1)
• Intensity is constant along a ray travelling
through space
• The Planck function (see next slide) is
expressed as intensity
+
Il
Il
• Radiative flux (Fl) is the net rate at which
energy having a wavelength between l and
l+l passes through unit area in a particular
direction (W/m m-2 sr-1)
• Think of it as adding up all the Il components
travelling in a particular direction
• For an isotropic radiation field Fl=0
F.Nimmo EART164 Spring 11
Black body basics
1. Planck function (intensity):
3
2h
1
B 
c
2
e
h
kT
1
Defined in terms of frequency or wavelength.
Upwards (half-hemisphere) flux is 2p B
2. Wavelength & frequency:  
3. Wien’s law: l max 
0 . 29
T
c
l
lmax in cm
e.g. Sun T=6000 K lmax=0.5 mm
Mars T=250 K lmax=12 mm
4. Stefan-Boltzmann law F 


B d    T
4
=5.7x10-8 in SI units
0
F.Nimmo EART164 Spring 11
Optical depth, absorption, opacity
I-I
z
I=-Ia z
a=absorption coefft. (kg-1 m2)
=density (kg m-3)
I = intensity
dI l  a l  I l dz
I
• The total absorption depends on  and a, and how they vary with z.
• The optical depth t is a dimensionless measure of the total
absorption over a distance h:
h
dt
t   a dz
= ar
0
dz
• You can show (how?) that I=I0 exp(-t)
• So the optical depth tells you how many factors of e the incident
light has been reduced by over the distance d.
• Large t = light mostly absorbed.
F.Nimmo EART164 Spring 11
Source function
• For a purely absorptive atmosphere, we have

1
dI l
a l  dz
 Il
• If the atmosphere is emitting as a black body
(and is in local thermal equilibrium) then we
dI l
1 dI l
get

a l  dz

dt l
 I l  Bl
• In the latter case, the changeover from the
incident light to the local source happens at
roughly t=1 (as expected)
F.Nimmo EART164 Spring 11
Absorption & opacity
dt
= ar
dz
• The absorption coefficient a is also called the opacity*
• You can think of it representing the surface area of absorbers of a
given mass
• Opacity depends strongly on temperature, phase (e.g. are clouds
present?) and composition, as well as wavelength
• For gases, opacity is often between 10-2 – 10-5 m2 kg-1 (higher at
high temperatures)
• For solid spheres of radius r and solid density s
3
3h
(This is the same
a=
t 
as Patrick’s equation)
4r rs
4 r s
• Example: Mars dust storms
*Really, the Rosseland mean opacity
because we’re integrating over all
wavelengths
Optical depth 1 would be
a smoggy day in LA
Here  is mass of solids/unit volume
h is total thickness
F.Nimmo EART164 Spring 11
Radiative Transfer for Non-Experts*
Black body radiation at a point is isotropic
But if there is a temperature gradient, there will still be a net flux
(upwards – downwards radiation)
For an atmosphere in equilibrium, the flux gradient must be zero
everywhere (otherwise it would be heating/cooling)
COLD
Net flux F
We can write the relationship
between the net flux and the
upwards-downwards radiation as
follows:
dB 
dt
HOT
See Lissauer and DePater 3.3.1

3
4p
F
Annoying
geometrical
factor
* That would be me
F.Nimmo EART164 Spring 11
Radiative Diffusion
• We can then derive (very useful!):
F (z)  
4p  T
3  z

1  B (T )
a
0

T
d
• If we assume that a is constant and cheat a
bit, we get
3
F (z)  
16  T  T
3  z
a
• Strictly speaking a is Rosseland mean opacity
• But this means we can treat radiation transfer as
a heat diffusion problem – big simplification
F.Nimmo EART164 Spring 11
Radiative Diffusion - Example
F (z)  
16  T  T
3  z
3
a
Earth’s mesosphere has a temperature
of about 230 K, a temperature
gradient of 3 K/km and an elevation
of about 70km.
Roughly what is the density?
What opacity would be required to
transmit the solar flux of ~103 Wm-2?
F.Nimmo EART164 Spring 11
Greenhouse effect
• “Two-stream” approximation – very like we
discussed in Week 2
• Atmosphere heated from below (no downwards flux
at top of atmosphere)
• This gives us two
Upwards flux F +2pB (T0)4pB(T0)
T0
Downwards flux F 2pB (T0)0
useful results we’ve
seen before:
Net flux F
T1
Net flux F
T0 
Upwards flux F +2pB (T1)
Surface Ts
2
1/ 4
T eq
3 

T (t )  T  1 + t 
2 

4
Downwards flux 2pB (T1)
1
4
0
It also means the effective depth
of radiation is at t =2/3
F.Nimmo EART164 Spring 11
Greenhouse effect
3 
4
4 
T (t )  T0  1 + t 
2 

4
T0 
1
2
4
T eq
A consequence of this model is that the surface is hotter than air
immediately above it. We can derive the surface temperature Ts:
Ts
4
3 

 T 1 + t s 
4 

4
eq
Earth
Mars
Teq (K)
255
217
T0 (K)
214
182
Ts (K)
288
220
Inferred t
0.84
0.08
Fraction transmitted
0.43
0.93
F.Nimmo EART164 Spring 11
Convection vs. Conduction
• Atmosphere can transfer heat depending on
opacity and temperature gradient
• Competition with convection . . .
dT
dz

3 T
16 T
4
e
3
a R
dT
dz

g
Cp
Whichever is smaller wins
-dT/dzad
Radiation dominates
(low optical depth)
-dT/dzrad
crit
 crit 
16
3
gT
3 a R Te C p
4
Does this equation make sense?
Convection dominates
(high optical depth)
F.Nimmo EART164 Spring 11
Radiative time constant
Atmospheric heat capacity (per m2):
C p H
Radiative flux:
 Te
Time constant:
Cp
4
( )T
P
g
F solar (1  A )
E.g. for Earth time constant is ~ 1 month
For Mars time constant is a few days
F.Nimmo EART164 Spring 11
Key Concepts
•
•
•
•
•
•
Black body radiation, Planck function, Wien’s law
Absorption, emission, opacity, optical depth
Intensity, flux
Radiative diffusion, convection vs. conduction
Greenhouse effect
Radiative time constant
F.Nimmo EART164 Spring 11
Key equations
dI l  a l  I l dz
Absorption:
dt
= ar
dz
Optical depth:
Greenhouse
effect:
Radiative
Diffusion:
3 

T (t )  T  1 + t 
2 

4
4
0
F (z)  
16  T  T
3  z
Cp
Rad. time constant:
T0 
1
2
1/ 4
T eq
3
a
( )T
P
g
F solar (1  A )
F.Nimmo EART164 Spring 11
End of lecture
F.Nimmo EART164 Spring 11
Simplified Structure
What does “optical depth” mean?
dl
I

dI 
Transmitted
dl
Absorbed
Scattered
dt
Emitted
dl
dI 
Simplest example with no emission
dt
  I  a  + j 
 a 
  I +
j
a
What are units of absorption?
F.Nimmo EART164 Spring 11
dI 
dt
  I +
j
a
LTE – emiss = absorb
dI 
dt
  I  + B (T ) 
F 
16  T
3 a
3
R
dB 
dt
Mathematical trickery!
dT
dz
Useful!
What are applications?
F.Nimmo EART164 Spring 11
Simplified Structure
What does “optical depth” mean?
4  TT 1 
3
dl

1
2
a  lT 1
4
Transmitted
8T1  dT
3
Absorbed+
Re-radiated
F   T1 
4
a
8T1  dT
dz
3
F   T1 
4
a
Scattering neglected
dz
 8T1 
3
dT
dt
t (z) 

T ( z )  T 1 +

2 

4
4
eq
What are units of absorption?Approx expression, actually ¾
What happens as tau-> 0?
F.Nimmo EART164 Spring 11