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
Line intensity

2. The basics - 1
Units
wavelength l (m), frequency n (Hz), wavenumber (m-1)
F flux density W m 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: mm
visible: mm
near-infrared: mm
Infrared / Longwave spectrum mm
C = x 108 m s-1

3. 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

4. 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

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

6. 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

7. 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

8. 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

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

10. 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 x Js
k is Boltzmann’s constant x JK-1
c is the speed of light in a vacuum m s-1

11. 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=2hc c2=hc/k x=c2/(lT) l5=c25 /(x5T5)
lmax Tmax = 2897 mm K
F = pB(T) = sT4

12. 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 mm
electronic energy (+vibr.+rot.): lines in the visible and UV

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. Approximations - 3
Mean line intensity
Mean half-width

20. 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

21. 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

22. 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

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

24. 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

25. 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 << Rayleigh scattering
aerosols < r < 10 mm
cloud particles 5 < r < 200 mm, rain drops and hail particles up to 1 cm

26. 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

27. 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

28. 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.