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 Line intensity

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

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

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

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

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

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

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

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

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 m s-1

The basic laws - 2 F = pB(T) = sT4 Wien’s law Stefan Boltzmann’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

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

The basic laws - 4 Line width In theory and 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

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

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

Approximations - 0 What is required in any RT scheme? Transmission function band model scaling and Curtis-Godson approximations correlated-k distribution Diffusivity approximation Scattering by particles

Approximations - 1 What is required to build a radiation transfer scheme for a GCM? 5 elements, the last, 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

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

Approximations - 3 Mean line intensity Mean half-width

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

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

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

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

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

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~10-4 mm 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

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

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 =-1 all energy is backscattered g = 0 equipartition between forward and backward spaces g = 1 all energy is in the forward space

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 spectral intervals of the 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 (no scattering), in each spectral intervals of the scheme.