Atmo II 80
Physics of the Atmosphere II
(4) Radiation Laws 2
Atmo II 81
Wien’s Displacement Law
Plank’s Radiation Law (Slides 36 – 41) already showed us, that the
Maximum of the spectral distribution of blackbody radiation is at long
wavelengths when temperatures are low (and vice versa). This has
already benn known before – as Wien's displacement law (Wilhelm Wien,
1893):
max
2 898 μm K

T
λmax is the wavelength, where the maximal radiation is emitted.
Inserting approximate values (of ~5800 K and ~290 K, respectively) gives:
Sun: λmax = 0.5 µm – Visible Light
Earth: λmax = 10 µm – Thermal Infrared
The Earth radiate – predominantly – in the infrared part of the spectrum.
Atmo II 82
Wien’s Displacement Law
We have already seen that the Stefan-Boltzmann law can be derived by
integrating the Plank’s radiation law (Slides 42 – 44). Also Wien's
displacement law follows from Plank’s law – now we need to differentiate
it (with respect to λ), and we get λmax by setting the first derivative = 0.

B
10 h c 
h c0

 1
exp
6


k B T 

2
0
1
2

 h c0
2hc 
h c0
 5 exp
 1  exp 
 
k B T 
 k B T
2
0
1


 h c0
h c0
 exp
 1  exp 
 
k B T 
 k B T
5
 h c0
 
2
k

 B T
  h c0
   
2
k

  B T

  0

Atmo II 83
Wien’s Displacement Law
Setting
h c0
x
k B T
with the solution
we get
x exp( x )
5
exp( x )  1
x  4.9651
h c0
6.62607 10 34 2.99792  108 m K
maxT 

Js
 23
k B xmax
1.38065  10 4.9651
s J
maxT  2.8978  10 3 m K
Atmo II 84
Kirchhoff’s Law
A black body , by definition, absorbs radiation at all wavelengths
completely. Real objects are never entirely “black” – the cannot absorb all
wavelengths completely, but show a wavelength-dependent
absorptivity ε(λ) (which is < 1).
According to Kirchhoff’s law (Gustav Kirchhoff, 1859) the Emission of a
body, Eλ (in thermodynamic equilibrium) is:
E  (,T )
 B (,T )
ε ( )
For a given wavelength and temperature, the ratio of the Emission and the
absorptivity equals the black body emission.
This shows also, that objects emit radiation in the same parts of the
spectrum in which they absorb radiation.
Atmo II 85
Kirchhoff’s Law
We rearrange Kirchhoff’s law and see:
E  (,T )  ε ( ) B (,T )
At a given temperature, real objects emit less radiation than a black body
(since ε < 1). Therefore we can regard ε(λ) also as emissivity. Quite often
you will thus find Kirchhoff’s law in the form:
Emissivity = Absorptivity
Important: it applies wavelength-dependent.
In the infrared all naturally occurring surfaces are – in very good
approximation – “black” – even snow! (which is – usually – not
black at all in the visible part of the spectrum).
For the Earth as a whole (in the IR): ε = 0.95
(„gray body“)
Atmo II 86
Radiation Balance
Terrestrial Radiation
Solar Radiation
Picture credit: NASA
At its (effective) “surface“, a planet will (usually) gain or lose energy only in
the form of radiation. In equilibrium we therefore get:
Incoming Radiation = Outgoing Radiation
Atmo II 87
Radiation Balance
We can us this, to build a (very simple!) zero-dimensional radiation balance
mode (note – here we regard Earth just as a point!). The Earth absorbs
shortwave solar radiation with its cross section (= area of a circle), but
emits (longwave) terrestrial radiation from its entire surface (= surface of a
sphere):
S0 1 A   R  ε T 4 R
2
E
4
Ef f
2
E
S0
4
1 A  ε TEf f
4
which gives the effective temperature of the Earth – which is –16 °C (!).
This is pretty far from Earth’s mean surface temperature of (meanwhile)
+15 °C.
What is wrong?
Atmo II 88
Infrared Active Gases
If we want to regard the „surface“ on slide 87 as the Earth’s surface, we
need to consider the influence of the Earth’s atmosphere – which is
(largely) transparent for solar radiation, but not for terrestrial radiation, since
it contains infrared active gases (pictures: C.D. Ahrens).
Atmo II 89
“Greenhouse Effect” (Basics)
Infrared active are (mainly) those gases with three or more atoms, which
show rotation-vibration bands in the infrared*: H2O, CO2, O3, N2O, CH4.
They are commonly termed “greenhouse gases” – but the term is not a
perfect choice (since the main reason, why a greenhouse is warmer that the
surrounding, is not the den “greenhouse effect”). “Greenhouse gases” also
emit infrared radiation, up and down. The part, which is emitted
downwards, warms the Earth’s surface.
With increasing temperature
the Earth’s surface emits more
IR-radiation (Stefan-Boltzmann
law), until an equilibrium
temperature is reached, where
the part of the IR-radiation, which
can leave the Earth’s atmosphere, equals the incoming
solar radiation.
Atmo II 90
“Greenhouse Effect” (Basics)
In our zero-dimensional model we can represent the influence of the infrared
active gases with the transmissivity in the infrared (τIR):
S0
1 A   IRε TEf4 f
4
With a value of 0.634 we get the mean surface temperature of +15 °C.
Without the selective absorption in the IR the Earth’s surface temperature
would be more than 30 °C lower (in his model world).
Anthropogenic CO2-Emissions enhance the “natural greenhouse effect”, bei
where water vapor – H2O dominates (!).
But in an atmosphere „without greenhouse gases“ there would not be snow
and clouds, the albedo would be less (A = 0.15) – an the means surface
temperature would by about –2°C (still pretty cold).
Atmo II 91
“Greenhouse Effect” (Basics)
As soon, as we look a bit
closer (later), things get
more complicated (NASA).
Atmo II 92
“Greenhouse Effect” (Basics)
More realistic energy balance (IPCC, 2007 after Kiel and Trenberth, 1997).
Atmo II 93
Longwave–Radiation
Net-Longwave Radiation = LWdown – LWup on the
Earth‘s surface. Absolute values but also annual
variations are surprisingly small, especially over the
ocean: Higher temperatures lead to more emitted
radiation (Stefan-Boltzmann) – but also to more water
vapor (Clausius-Clapeyron) – and therefore more „back
radiation“.
Atmo II 94
Radiation Balance – Annual Cycle
Net-Shortwave Radiation
Net-Radiation
Net-Longwave Radiation
Net-Shortwave Radiation = SWdown – SWup
Net-Longwave Radiation = LWdown – LWup
Net-Radiation = Net-SW – Net-LW
Atmo II 95
Radiation and Temperature
Net-Radiation
The surface temperature (below)
follows (roughly) the net-radiation
(left). But note the (way) less
pronounced meridional movement of
the temperature maximum, due to the
thermal inertia of the oceans.
Surface Temperature
The increasing distance of the
isotherms in the eastern parts of the
ocean basins, particularly in the North
Atlantic, are caused by ocean
currents (North Atlantic Current +
Canary Current).