di λ ds = ρk λi λ B λ (T ) + ρk λ dz' )= B λ (T(z'))e I λ (z TOA + k λ Longwave radiative transfer Longwave radiative transfer

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1 Radiative transfer applied to the atmosphere: Longwave radiation z In the longwave spectrum (> µm) we have to take into account both absorption and emission, as the atmosphere and Earth surface with temperatures of K emit in this spectral range according to the Planck law. cattering is negligible. ε λ ρdz = k λ ρdz (Kirchhof) I λ I λ ρk λ dz=i λ (1-ρk λ dz) dz k λ, ρ di=-i λ ρk λ dz The source function of an atmospheric layer is k λ (T) I λ (longwave) The radiative transfer equation for the longwave radiation thus becomes: ε λ ρdz = k λ ρdz di λ ds = ρk λi λ + ρk λ (T ) Equation of chwarzschild di λ dz = k λρ( I λ ) Equation of chwarzschild olution of chwarzschild equation: z' (T s ) k λ (T(z')) dz' Emission/absorption coefficient k λ I λ ( )= (T s )e 0 k λ ρdz z' + k λ (T(z'))e 0 k λ ρdz dz' trongly wavelength dependent The intensity at the top of atmosphere equals the surface radiation attenuated by the transmittance of the atmosphere, plus the sum of all contributions of atmospheric radiation from all vertical layers dz each contribution attenuated by the transmittance from that level z to the top of atmosphere

2 Atmospheric window Little absorption by any of the atmospheric gases between 8 and 1 µm permits substantial longwave radiation loss to space c.f. maximum emission of Earth surface around 10 µm (Wiens law) In atmospheric window the earth-atmosphere system radiates very close to the earth's surface temperature Depending on wavelength, longwave radiation is emitted from different altitudes, and therefore from different temperature levels pectrum of longwave radiation emitted by the Earth as measured from space pectral emission with different levels of CO 2 Radiation fluxes emitted to space at three different wavelengths and for the temperature profile in the left panel. Opaque regions of the atmosphere are shown in gray shading.

3 Idealised case: upward longwave radiation from the Earth s surface does not get absorbed by any gases in the atmosphere. The longwave radiation at the top of the atmosphere would look like this: Area under curve (integral over wavelenght) equals 20 Wm -2 What would happen if the atmosphere allowed radiation only through the atmospheric window and everywhere else the transmittance was zero Current Earth surface temperature (15 C) would only emit 11 Wm -2 to space in this situation urface temperature heats up until it fully compensates for 20 Wm -2 obtained by the solar absorption Ø T surface of 323 K (50 C) required More close to reality: Transmission of radiation also outside the atmospheric window Increasing greenhouse effect => emission of radiation from different levels in the atmosphere outside the atmospheric window: =98 Wm -2 Plus the 11 Wm -2 from the atmospheric window = 239 Wm -2, in balance with shortwave absorption Increasing CO 2 in the atmosphere results in: decreasing outgoing radiation at the TOA (as main emission level is moved upward into colder regions) increasing downward longwave radiation at the surface (as main emission level is moved downward into warmer regions)

4 Increasing greenhouse effect Why is interior Antarctica not warming? Troposphere-surface system has to heat up to regain balance with shortwave absorption at the TOA Why is interior Antarctica not warming? Why is interior Antarctica not warming? Model experiments with x CO2 Prerequisit for greenhouse warming: Atmosphere higher up is cooler than Earth surface. Not always fulfilled under specific conditions of interior Antarctica (cool surfaces and temperature inversion in atmosphere). Emission to space FTOA: εatm= atmospheric emission/absorption More CO2 => εatm increases => if Tsurf > Tatm => FTOA decreases > warming => if Tsurf < Tatm => FTOA increases > cooling More LW emission to space Less LW emission to space Most places of the Earth : Tsurf > Tatm => More CO2 > warming Full study chmithüsen et al on Course Webpage Full study chmithüsen et al on Course Webpage

5 imple model of the greenhouse effect How much longwave radiation is trapped by the greenhouse effect? Earth surface emits with σt s =390Wm -2 (using T s =288K) Planet emits with effective Temperature σt eff = (1-A)/= 20 Wm -2 Amount of longwave radiation trapped by atmosphere: 150 Wm -2 Total energy in one year implifying assumptions: Atmosphere consists of only one layer with temperature T e Atmosphere completely transparent for shortwave radiation Earth surface has a emissivity of one (blackbody) and temperature T s Atmosphere lower (but non-zero) emissivity ε dependent on GHGs No consideration of non-radiative energy balance components such as sensible and latent heat (convection) Courtesy M.Mann Penn tate University imple model of the greenhouse effect Implications ε describes emission as well as absorption capacity of the atmosphere (>Kirchhoff Law) dependent on greenhouse gases ε =0: no greenhouse effect at all ε =1: atmosphere perfect absorber of longwave radiation (perfect greenhouse effect) True greenhouse effect somewhere in between (0 < ε <1) Courtesy M.Mann Penn tate University Balancing incoming and outgoing radiation from the atmospheric layer gives (simplifying assumption: solar radiation is not absorbed in the atmosphere): εσt = 2εσT e T e = T 2 (1) Balancing incoming and outgoing radiation at the surface gives: ( 1 A) Replacing T e by T in (2) using (1) and resolving for T : T + σε 2 T imple model of the greenhouse effect +σε T e = σt = σ = σt => T (2) => σt = ( 1 ε / 2) ( 1 ε / 2) σ ( 1 ε / 2) =

6 imple model of the greenhouse effect imple model of the greenhouse effect εσt = 2εσT e T e = T 2 T e = T 2 1/ (1) T = (1 A) σ ε ( 1 / 2) Examples ff: Temperature of the atmosphere T e T s = 288 K => T e = 22 K. This is somewhat lower than the effective radiating temperature T eff = 255 K, since not all radiation that escapes to space is emitted from the atmosphere, but partly from the warmer surface Examples: A = 0.3, = 1361 Wm -2 ε = 0 (no greenhouse effect): T = 255 K = -18 C. Too cold! ε = 1 (a perfectly longwave absorbing atmosphere), T = 303 K = 30 C. Too warm! ε = 0.77 (i.e., the atmosphere absorbs 77% of the longwave radiation incident upon it) : T = 288 K = 15 C. Just right! Top of atmosphere balance: (1 A) = εσt e + (1 ε)σt (1-ε)σT s

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