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ECMWFRadiation: Basic concepts and Approximations 1
The basics - 0
DefinitionsThe Radiative Transfer Equation (RTE)
The relevant lawsPlanck’sWiens’sStefan-BoltzmannKirchhoff’s
A bit of useful spectroscopyLine widthLine intensity
ECMWFRadiation: Basic concepts and Approximations 2
The basics - 1
Unitswavelength (m), frequency (Hz), wavenumber (m-1)
F flux density W m-2 flux per unit area, flux or irradianceL specific intensity W m-2 sr-1 flux per unit area into unit solid, radiance
Solar / Shortwave spectrumultraviolet: 0.2 - 0.4 mvisible: 0.4 - 0.7 mnear-infrared: 0.7 - 4.0 m
Infrared / Longwave spectrum4 - 100 m
vc /1/ C = 2.99793 x 108 m s-1
v
ECMWFRadiation: Basic concepts and Approximations 3
The basics - 2
The Radiative Transfer Equation (RTE)
For GCM applications, no polarization effectstationarity (no explicit dependence on time)plane-parallel (no sphericity effect)
Sources and sinks:ExtinctionEmissionScattering
ECMWFRadiation: Basic concepts and Approximations 4
The basics - RTE 1
ExtinctionRadiance L(z, entering the cylinder at one end is
extinguished within the volume (negative increment)
,ext is the monochromatic extinction coefficient (m-1)
d is the solid angle differentialdl the lengthda the area differential
dadldzLzdQ extext ),,(),,( ,,
ECMWFRadiation: Basic concepts and Approximations 5
The basics - RTE 2
Emission
,abs is the monochromatic absorption coefficient (m-1)B(T) is the monochromatic Planck function
dadldzTBzdQ absemis )]([),,( ,,
ECMWFRadiation: Basic concepts and Approximations 6
The basics - RTE 3
Scatteringchange of radiative energy in the volume caused by scattering of
radiation from direction (’,’) into direction (,)
,scat is the monochromatic scattering coefficient
d’ is the solid angle differential of the incoming beamP(z,’’) is the normalized phase function, I.e., the probability
for a photon incoming from direction (’,’) to be scattered in direction (,), with
')',',(4
)',',,,(),,( ,,
ddadldzL
zPzdQ scatscat
14
1
Pd
ECMWFRadiation: Basic concepts and Approximations 7
The basics - RTE 4
Since scattered radiation may originate from any direction, need to integration over all possible (’,’)
The direct unscattered solar beam is generally considered separately
E is the specific intensity of the incident solar radiation (o,o) is the direction of incidence at ToAo is the cosine of the solar zenith angle is the optical thickness of the air above z
'4
)',',,,()',',(),,(
'
,
d
zPzLdadldzdQ
w
scat
dadldE
zPzdQ o
oo
o scat )exp(4
),,,,(),,(
0,
ECMWFRadiation: Basic concepts and Approximations 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
dzZtoa
z
ext ,
dzdz
dl cos
dzd ext,
ext
scat
,
,
only absorption 0 only scattering 1
ECMWFRadiation: Basic concepts and Approximations 9
The basics - RTE 6
The most general expression of the radiative transfer equation is
))(()1(
4
),,,,()exp(
'4
)',',,,()',',('
),,(
2
0
1
1
TB
PE
dP
Ld
Ld
dL
v
ooo
ECMWFRadiation: Basic concepts and Approximations 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:
hE h is Planck’s constant 6.626 x 10-34 Js
1
5
2
1exp2
)(
kT
hchcTB
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
vTBTBTB v )()()(
ECMWFRadiation: Basic concepts and Approximations 11
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 absorptivity
0max dT
dB c1=2hc2 c2=hc/k x=c2/(T) 5=c25 /(x5T5)
0)1)(exp(
ln2
551
xc
xTc
dx
d
max Tmax = 2897 m K
dxx
x
ch
TkdTBTB
0
3
23
44
0 1)exp(
2)()(
F = B(T) = T4
The basic laws - 2
ECMWFRadiation: Basic concepts and Approximations 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 E=h steps rotational energy: lines in the far infrared > 20mvibrational energy (+rotational): lines in the 1 - 20 melectronic energy (+vibr.+rot.): lines in the visible and UV
ECMWFRadiation: Basic concepts and Approximations 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
2
05.0
exp)(
)(DD
DD
Sk
hE /
)2
exp(2
)(25.0
KT
mv
kT
mdvvP
5.020
m
KT
cD
ECMWFRadiation: Basic concepts and Approximations 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
220 )(
)(L
LL
Sk
5.00
00
T
T
P
PLL
ECMWFRadiation: Basic concepts and Approximations 15
Line intensity
0
00
11exp
TTk
E
T
TSS
x
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
The basic laws - 4
ECMWFRadiation: Basic concepts and Approximations 16
Approximations - 0
What is required in any RT scheme?
Transmission functionband modelscaling and Curtis-Godson approximationscorrelated-k distribution
Diffusivity approximation
Scattering by particles
ECMWFRadiation: Basic concepts and Approximations 17
Approximations - 1What 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
ECMWFRadiation: Basic concepts and Approximations 18
Approximations - 2
Band models of the transmission function over a spectral interval of width
Goody
Malkmus
2
1
)1(exp
SaSa
]1)1[(
2exp 2
1
Sa
S are the mean intensity and the mean half-width of the N lines within with mean distance between lines
ECMWFRadiation: Basic concepts and Approximations 19
Approximations - 3
Mean line intensity
Mean half-width
N
iSN
S1
1
2
1
2
1
)(11
N
iiSNS
ECMWFRadiation: Basic concepts and Approximations 20
Approximations - 4
In order to incorporate the effect of the variations of the ,x coefficients with temperature T and pressure p
Scaling approximation
a
dazTzpzTSf0
'))](),(()),(([
errr aTpTSf )],(),([
')'()'(
0
daT
aT
p
apa
y
r
a x
re
The effective amount of absorbercan be computed with x,y coefficients defined spectrally orover the whole spectrum
ECMWFRadiation: Basic concepts and Approximations 21
Approximations - 5
2-parameter or Curtis-Godson approximation
)()()( TTSTS r
)(),()(),()( TpTTSpTTS rrrr
N
ri
N
i
TS
TST
1
1
)(
)()(
2
1
2
1
2
1
2
1
),()(
),()(
)(
N
rriri
N
rii
pTTS
pTTS
T
All these parameters can be computed from the information, i.e., the Si , i ,included in spectroscopic database like HITRAN
ECMWFRadiation: Basic concepts and Approximations 22
Approximations - 6
Correlated-k distribution (in this part ki=,abs)ki, the absorption coefficient shows extreme spectral variation.
Computational efficiency can be improved by replacing the integration over 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 j which have absorption coefficient within a range k and ki+ki
The cumulative frequency distribution increments define the fraction of the interval for which kv is between ki and ki+ki
),(1
)(12
iii
M
j i
ji kkkW
kkf
iii kkfg )(
ECMWFRadiation: Basic concepts and Approximations 23
Approximations - 7
The transmission function, over an interval [1,2], can therefore be equivalently written as
2
1
)exp(1
)(12
dakaT
ii
N
i
i kakkfaT
)exp()()(1
N
i
ii gakaT1
)exp()(
0
)exp()()( dkkakfaT
1
0
))(exp()( dgagkaT
ECMWFRadiation: Basic concepts and Approximations 24
Approximations - 8
Diffusivity factora flux is obtained by integrating the radiance L over the anglewith the transmission in the form
the exact solution involves the exponential integral function of order 3
)/exp( x
1
0
),(2 dyLF
1
3 )exp()exp()( rxdyyxyxE n
where r ~ 1.66 is the diffusivity factor
ECMWFRadiation: Basic concepts and Approximations 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 = 2 rmolecules r~10-4 m << 1 Rayleigh scattering
aerosols 0.01 < r < 10 m
cloud particles 5 < r < 200 m, rain drops and hail particles up to 1 cm
4/ '
'*
, absm
immm
ECMWFRadiation: Basic concepts and Approximations 26
Scattering by particles - 2
Rayleigh scattering
size of air molecules r << wavelength of radiation, i.e., <<1
phase function
conservative
completely symmetric: asymmetry factor g=0 probability of scattering ~ density of air )~ 1 / 4
)cos1(4
3 2 RP
1
ECMWFRadiation: Basic concepts and Approximations 27
Scattering by particles - 3
Mie scattering r ~ 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)
)'( )( )12()',,( lll PPlP
'21
1)',,(
2
2
gg
gP
1
1
)',,(2
1 dPgg =-1 all energy is backscatteredg = 0 equipartition between forward and backward spacesg = 1 all energy is in the forward space
ECMWFRadiation: Basic concepts and Approximations 28
Scattering by particles - 4
Mie scattering
aerosols: development in Legendre polynomials
clouds particles
2
ext
In the ECMWF model, optical properties for liquid and ice clouds and aerosols are represented through optical thickness, single scattering albedo, and asymmetryfactor, 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.