(4) Radiation Laws 2

Physics of the Atmosphere II

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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):

λmax is the wavelength, where the maximal radiation is emitted.

T

Kμm8982max

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Inserting approximate values (of ~5800 K and ~290 K, respectively) gives:

Sun: λmax = 0.5 µm – Visible LightEarth: λmax = 10 µm – Thermal InfraredThe Earth radiate – predominantly – in the infrared part of the spectrum.

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.

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0exp1exp2

1exp10

200

2

05

20

1

06

20

Tk

ch

Tk

ch

Tk

chch

Tk

chchB

BBB

B

Tk

ch

Tk

ch

Tk

ch

BBB200

1

0 exp1exp5

Wien’s Displacement Law

Setting we get

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1)exp(

)exp(5

x

xx

Tk

chx

B0

with the solution 4.9651x

J

K

s

mJs

4.9651101.38065

102.99792106.6260723

834

max

0max

xk

chT

B

Km102.8978 3max

T

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:

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.

)()(

)(TB

TE,

ε

,

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We rearrange Kirchhoff’s law and see:

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.

)()()( TBTE ,ε,

Kirchhoff’s Law

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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“)

Radiation Balance

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

Solar Radiation

Terrestrial Radiation

Picture credit: NASA

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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):

2E

4Ef f

2E 4ε1 RTRAS 0

4Ef fε1

4TA

S 0

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?

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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).

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“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 atmo-sphere, equals the incoming solar radiation.

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In our zero-dimensional model we can represent the influence of the infrared active gases with the transmissivity in the infrared (τIR):

4Ef fIRε1

4TA

S 0

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).

“Greenhouse Effect” (Basics)

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“Greenhouse Effect” (Basics)

As soon, as we look a bit closer (later), things get more complicated (NASA).

More realistic energy balance (IPCC, 2007 after Kiel and Trenberth, 1997).

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“Greenhouse Effect” (Basics)

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“.

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Net-Shortwave Radiation

Net-Radiation

Net-Longwave Radiation

Net-Shortwave Radiation = SWdown – SWup

Net-Longwave Radiation = LWdown – LWup

Net-Radiation = Net-SW – Net-LW

Radiation Balance – Annual Cycle

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Net-Radiation

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).

Radiation and Temperature

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.

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