# (4) Radiation Laws 2 Physics of the Atmosphere II Atmo II 80.

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(4) Radiation Laws 2 Physics of the Atmosphere II Atmo II 80

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): λ max is the wavelength, where the maximal radiation is emitted. Atmo II 81 Inserting approximate values (of ~5800 K and ~290 K, respectively) gives: Sun: λ max = 0.5 µm – Visible Light Earth: λ max = 10 µm – Thermal Infrared The 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 λ max by setting the first derivative = 0. Atmo II 82

Wien’s Displacement Law Setting we get Atmo II 83 with the solution

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, E λ (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. Atmo II 84

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. Kirchhoff’s Law Atmo II 85 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 Atmo II 86

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): 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 87

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 88

“Greenhouse Effect” (Basics) Infrared active are (mainly) those gases with three or more atoms, which show rotation-vibration bands in the infrared*: H 2 O, CO 2, O 3, N 2 O, CH 4. 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. Atmo II 89

In our zero-dimensional model we can represent the influence of the infrared active gases with the transmissivity in the infrared ( τ IR ): 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 CO 2 -Emissions enhance the “natural greenhouse effect”, bei where water vapor – H 2 O 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) Atmo II 90

Atmo II 91 “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). Atmo II 92 “Greenhouse Effect” (Basics)

Longwave–Radiation Net-Longwave Radiation = LW down – LW up 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 93