The basics - 0

1. The basics - 0 • Definitions • The Radiative Transfer Equation (RTE) • The relevant laws • Planck’s • Wiens’s • Stefan-Boltzmann • Kirchhoff’s • A bit of useful spectroscopy • Line width (Lorentz, Doppler) • Line intensity

2. L. Boltzmann J. Stefan M. Planck W. Wien H.A. Lorentz C. Doppler G. Kirchhoff

3. The basics - 1 • Units • wavelength l (m), frequency n (Hz), wavenumber (m-1) • F flux density W m-2 flux per unit area, flux or irradiance • L specific intensity W m-2 sr-1 flux per unit area into unit solid, radiance • Solar / Shortwave spectrum • ultraviolet: 0.2 - 0.4 mm • visible: 0.4 - 0.7 mm • near-infrared: 0.7 - 4.0 mm • Infrared / Longwave spectrum • 4 - 100 mm C = 2.99793 x 108 m s-1

5. The basics - 2 • The Radiative Transfer Equation (RTE) • For GCM applications, • no polarization effect • stationarity (no explicit dependence on time) • plane-parallel (no sphericity effect) • Sources and sinks: • Extinction • Emission • Scattering

7. The basics - RTE 1 • Extinction • Radiance Ln(z, q, f) entering the cylinder at one end is extinguished within the volume (negative increment) • bn,ext is the monochromatic extinction coefficient (m-1) • dw is the solid angle differential • dl the length • da the area differential

8. The basics - RTE 2 • Emission • b n,abs is the monochromatic absorption coefficient (m-1) • Bn(T) is the monochromatic Planck function

9. The basics - RTE 3 • Scattering • change of radiative energy in the volume caused by scattering of radiation from direction (q’,f’) into direction (q,f) • b n,scat is the monochromatic scattering coefficient • dw’ is the solid angle differential of the incoming beam • Pn(z,q,f,q’,f’) is the normalized phase function, I.e., the probability for a photon incoming from direction (q’,f’) to be scattered in direction (q,f), with

10. The basics - RTE 4 • Since scattered radiation may originate from any direction, need to integration over all possible (q’,f’) • The direct unscattered solar beam is generally considered separately • Eon is the specific intensity of the incident solar radiation • (qo,fo) is the direction of incidence at ToA • mo is the cosine of the solar zenith angle • dn is the optical thickness of the air above z

11. The basics - RTE 5 • The optical thickness is given by • The total change in radiative energy in the cylinder is the sum, and after replacing dl by the geometrical relation • considering that • and introducing the single scattering albedo

12. The basics - RTE 6 • The most general expression of the radiative transfer equation is

13. The basic laws - 1 • Planck’s law • for one atomic oscillator, change of energy state is quantized • for a large sample, Boltzmann statistics (statistical mechanics) • NB: h is Planck’s constant 6.626 x 10-34 Js k is Boltzmann’s constant 1.381 x 10-23 JK-1 c is the speed of light in a vacuum 2.9979 x 108 m s-1

15. The basic laws - 2 • Wien’s law • extremes of the Planck function are defined by • Stefan Boltzmann’s law • Kirchhoff’s law: in thermodynamic equilibrium, i.e., up to ~50-70 km depending on gases emissivity el = absorptivity al c1=2hc2 c2=hc/k x=c2/(lT) l5=c25 /(x5T5) lmax Tmax = 2897 mm K F = pB(T) = sT4

16. The basic laws - 3 • Spectral behaviour of the emission/absorption processes • Planck function has a continuous spectrum at all temperatures • Absorption by gases is an interaction between molecules and photons and obeys quantum mechanics • kinetic energy: not quantized ~ kT/2 • quantized:changes in levels of energy occur by DE=h Dn steps • rotational energy: lines in the far infrared l > 20mm • vibrational energy (+rotational): lines in the 1 - 20 mm • electronic energy (+vibr.+rot.): lines in the visible and UV

18. The basic laws - 4 • Line width • In theory Dand lines are monochromatic • Actually, lines are of finite width, due to natural broadening (Heisenberg’s principle) • Doppler broadening due to the thermal agitation of molecules within the gas: from a Maxwell-Boltzmann probability distribution of the velocity • the absorption coefficient of such a broadened Doppler line is • with

19. The basic laws - 5 • Line width • Pressure broadening (Lorentz broadening) due to collisions between the molecules, which modify their energy levels. The resulting absorption coefficient is • with the half-width proportional to the frequency of collisions

21. The basic laws - 4 • Line intensity E is the energy of the lower state of the transition x is an exponent depending on the shape of the molecule 1 for CO2, 3/2 for H2O, 5/2 for O3 T0 is the reference temperature at which the line intensities are known

22. Approximations - 0 • What is required in any RT scheme? • Transmission function • band model • scaling and Curtis-Godson approximations • correlated-k distribution (used in the present McRad scheme) • Diffusivity approximation • Scattering by particles

23. Approximations - 1 • What is required to build a radiation transfer scheme for a GCM? • 5 elements, the last four, in principle in any order: • a formal solution of the radiation transfer equation • an integration over the vertical, taking into account the variations of the radiative parameters with the vertical coordinate • an integration over the angle, to go from a radiance to a flux • an integration over the spectrum, to go from monochromatic to the considered spectral domain • a differentiation of the total flux w.r.t. the vertical coordinate to get a profile of heating rate

24. Approximations - 2 • Band models of the transmission function over a spectral interval of width Dn • Goody • Malkmus are the mean intensity and the mean half-width of the N lines within Dn, with mean distance between lines d

25. Approximations - 3 • Mean line intensity • Mean half-width All these parameters can be computed from the information, i.e., the Si , ai , included in spectroscopic database like HITRAN

26. Approximations - 4 • In order to incorporate the effect of the variations of the b n,x coefficients with temperature T and pressure p • Scaling approximation The effective amount of absorber can be computed with x,y coefficients defined spectrally or over the whole spectrum

27. Approximations - 5 • 2-parameter or Curtis-Godson approximation All these parameters can be computed from the information, i.e., the Si , ai , included in spectroscopic database like HITRAN

28. Approximations - 6 • Correlated-k distribution (in this part ki=bn,abs) • ki, the absorption coefficient shows extreme spectral variation. • Computational efficiency can be improved by replacing the integration over l with a reordered grouping of spectral intervals with similar ki strength. • The frequency distribution is obtained directly from the absorption coefficient spectrum by binning and summing intervals Dnj which have absorption coefficient within a range ki and ki+Dki • The cumulative frequency distribution increments define the fraction of the interval for which kv is between ki and ki+Dki Lacis, A.A. and V. Oinas, 1991: J.Geophys.Res., 96D, 9027-9063.

31. Approximations - 7 • The transmission function, over an interval [n1,n2], can therefore be equivalently written as

32. Approximations - 8 • Diffusivity factor • a flux is obtained by integrating the radiance L over the angle • with the transmission in the form • the exact solution involves the exponential integral function of order 3 where r ~ 1.66 is the diffusivity factor r=1/cosqwith q ~ 53 degrees

33. Scattering by particles - 1 • Scattering efficiency depends on size r, geometrical shape, and the real part of its refractive index, whereas the absorption efficiency depends on the imaginary part • Intensity of scattering depends on Mie parameter a = 2 p r / l • molecules r~10-4mm a << 1 Rayleigh scattering • aerosols 0.01 < r < 10 mm • cloud particles 5 < r < 200 mm, rain drops and hail particles up to 1 cm

34. Scattering by particles - 2 • Rayleigh scattering • size of air molecules r << l wavelength of radiation, i.e., a <<1 • phase function • conservative • completely symmetric: asymmetry factor g=0 • probability of scattering ~ density of air bl(a)~ 1 / l4 J.W. Strutt, Lord Rayleigh

36. Scattering by particles - 3 • Mie scattering r ~ l • phase function developed into Legendre polynomials • for flux computation, only a few terms are required or some analytic formula as Henyey-Greenstein function can be applied • with g, the asymmetry factor (1st moment of the expansion) G. Mie g =-1 all energy is backscattered g = 0 equipartition between forward and backward spaces g = 1 all energy is in the forward space

38. Various drop size distributions for standard model clouds Total extinction coefficient

39. Asymmetry factor g Single scattering albedo w

40. Scattering by particles - 4 • Mie scattering • aerosols: development in Legendre polynomials • clouds particles In the ECMWF model, optical properties for liquid and ice clouds and aerosols are represented through optical thickness, single scattering albedo, and asymmetry factor, defined for each of the (6)/14 spectral intervals of the (old)/RRTM-SW scheme and each of the 16 spectral intervals of the RRTM-LW scheme. For liquid and ice clouds, optical properties are linked to an effective particle size, whereas for aerosols integration over the size distribution is actually included. In the LW, only total absorption coefficients are finally considered (scattering only approximated), in each spectral intervals of the scheme.