ecmwf radiation: basic concepts and approximations 1 the basics - 0 definitions the radiative...

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


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


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


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 ),,(),,( ,,


The basics - RTE 2

Emission

,abs is the monochromatic absorption coefficient (m-1)B(T) is the monochromatic Planck function

dadldzTBzdQ absemis )]([),,( ,,


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


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,


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


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


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 )()()(


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


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


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


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


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


Approximations - 0

What is required in any RT scheme?

Transmission functionband modelscaling and Curtis-Godson approximationscorrelated-k distribution

Diffusivity approximation

Scattering by particles


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


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


Approximations - 3

Mean line intensity

Mean half-width

N

iSN

S1

1

2

1

2

1

)(11

N

iiSNS


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


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


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 )(


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


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


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



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



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



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.

ecmwf radiation: basic concepts and approximations 1 the basics - 0 definitions the radiative...

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